Model theory of pseudoﬁnite structures

Model theory of pseudofinite structures

Darío García

University of Leeds

School of Mathematics

Institute for Research in Fundamental Sciences - IPM

Tehran, Iran

October 23-29, 2016

2

Contents

Introduction 5

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1 Basic concepts, examples and results. 9

1.1 Ultraproducts of finite structures . . . . . . . . . . . . . . . . . . . . . . . 91.2 Pseudofinite structures and theories . . . . . . . . . . . . . . . . . . . . . . 151.3 Examples of pseudofinite structures and theories . . . . . . . . . . . . . . . 17

2 Pseudofinite dimensions, forking and simplicity. 23

2.1 Measures and dimension in ultraproducts of finite structures . . . . . . . . 232.2 The pseudofinite dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Fine pseudofinite dimension and forking . . . . . . . . . . . . . . . . . . . 32

2.3.1 Dividing and drop of pseudofinite dimension . . . . . . . . . . . . . 322.3.2 Conditions on the fine pseudofinite dimension . . . . . . . . . . . . 36

2.4 Forking independence and δ-independence . . . . . . . . . . . . . . . . . . 392.4.1 Simplicity and forking . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5 Fine pseudofinite dimension and stability . . . . . . . . . . . . . . . . . . . 46

3 Applications of pseudofinite structures to combinatorics 49

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Furstenberg’s correspondence principle . . . . . . . . . . . . . . . . . . . . 513.3 Szemerédi’s Regularity Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 523.4 The Erdös-Hajnal property for stable graphs. . . . . . . . . . . . . . . . . 57

4 Strongly minimal pseudofinite structures 65

4.1 Strongly minimal structures . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Strongly minimal pseudofinite structures . . . . . . . . . . . . . . . . . . . 77

4.2.1 Unimoduarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

References 80

3

4 CONTENTS

Introduction

First order model theory on infinite structures is a well–developed field of abstract math-ematics with powerful methods and results an whose subject is to study the interactionbetween mathematical structures such as groups, fields, graphs, and the formal languagesused to describe them.

Throughout the history of model theory the subject have been fueled and redefined byits applications, especially in other areas of mathematics. Early examples of these appli-cations are Tarski’s elimination of quantifiers for real closed fields and Ax-Kochen/Ershovmode theory of henselian fields. More recently, connections with algebraic and diophan-tine geometry have been found and used to prove the Mordell-Lang conjecture for functionfields (Hrushovski) and to advance towards the proof of the André-Oort conjecture forCn (due to Pila) by using tools from o-minimality to find estimates of rational points indefinable sets.

Most of the applications of model theory to other areas in mathematics come in twostages: first by identify-ing abstract (often combinatorial) properties of first-order the-ories that make them more tractable or “tame” (such as stability, simplicity, NIP, andmore recently rosiness and NTP2), and second when we realize that theories of math-ematically meaningful structures satisfy those properties. The leading idea behind themost recent applications from model theory to other areas has been the slogan proposedby Hrushovski: “model theory is the geography of tame mathematics” (see [12], page 38),where model-theorists use informally the terms “tame” or “wild” to distinguish betweenhaving desirable or undesirable model-theoretic behavior.

In contrast, Finite Model Theory - the specialization of model theory to the studyfinite structures - has very different methods, and usually refers to a field of mathemat-ics which has more to do with computer science than to classical mathematical structures.

The fundamental theorem of ultraproducts is due to Jerzy Łoś, and provides a trans-ference principle between the finite structures and their limits. Roughly speaking, Łoś’Theorem states that a formula is true in the ultraproduct M of the structures 〈Mn : n ∈ N〉if and only if it is true for “almost every” Mn.

When applied to ultraproducts of finite structures, Łoś’ theorem presents an inter-esting duality between the finite structures and the infinite structures. We start with afamily of finite structures and produce infinite first-order structure with the same prop-

5

6 CONTENTS

erties. This kind of finite/infinite connection can sometimes be used to prove qualitativeproperties of large finite structures using the powerful known methods and results comingfrom infinite model theory, and in the other direction, quantitative properties in the finitestructures often induced desirable qualitative properties in their ultraproducts.

The idea is that the counting measure on a class of finite structures can be lifted usingŁoś’ theorem to give notions of dimension and measure on their ultraproduct. This allowsideas from geometric model theory to be used in the context of pseudofinite theories, andpotentially we can prove results in finite combinatorics (of graphs, groups, fields, etc) bystudying the corresponding properties in the ultraproducts. This approach was used byHrushovski in [14], but was better explored in his striking papers [15] and [16], wherehe applies ideas from geometric model theory to additive combinatorics, locally compactgroups and linear approximate subgroups.

Goldbring and Towsner developed in [10] the Approximate Measure Logic, a logicalframework that serves as a formalization of connections between finitary combinatoricsand diagonalization arguments in measure theory or ergodic theory that have appeared invarious places throughout the literature. Using AML- structures, Goldbring and Towsnergave proofs of the Furstenberg’s correspondence principle, Szemerédi’s Regularity Lemma,the triangle removal lemma, and Szemerédi’s Theorem: every subset of the integers withpositive density contains arbitrarily long arithmetic progressions.

There is a variety of possible questions about what is the relationship between thedifferent concepts in model theory (stability, NIP, simplicity, geometries coming fromindependence relations, etc) once the assumption of pseudofiniteness is added, and howthese classical model-theoretic properties on the ultraproducts of a class of finite structuresreflect on quantitative properties for the definable sets along the class.

Outline

The purpose of this series of lectures is to present the main topics about the model theoryof pseudofinite structures, together with some recent developments and applications tocombinatorics.

On the model-theoretic perspective, we will review the so-called “pseudofinite dimen-sion” (cf. [16]) and its relationship with the forking and model-theoretic dividing lines inpseudofinite structures as presented in [8]. We will also study folklore results appearingin [24] where some counting arguments on pseudofinite structures and the existence ofZilber polynomials are shown to imply that every strongly minimal pseudofinite structureis locally modular.

On the applications to combinatorics we will see a proof of Szemerédi’s RegularityLemma using ultraproducts of finite structures, (cf. [10]) as well as some improvementsof this result under the assumption of stability due to Malliaris and Shelah. This last

CONTENTS 7

approach was used by Chernikov and Starchenko in [6] to obtain some progress in thestudy of the Erdös-Hajnal conjecture.

Acknowledgements

The idea of these notes started by recollecting material for my lecture course Model Theoryof pseudofinite structures at the Institute for Research in Fundamental Sciences (IPM),in Tehran, Iran during October 24 - 29 2016.

I would like to thank the IPM members for their invitation and support, especiallyto Zaniar Ghadernezhad and Massoud Pourmahdian for making my visit possible and fortheir hospitality during my time in Tehran. I am also very grateful to the participants ofthe course for their attention and the very interesting questions and discussions proposedduring the lectures.

8 CONTENTS

Lecture 1

Basic concepts, examples and results.

In this chapter we will study the main definitions and examples in the study of L-structuresM which are pseudofinite, that is, L-structures which satisfy every first order sentencewhich hold for all finite L-structures. As we will see in Proposition 1.2.3, being pseudofi-nite is equivalent to being elementary equivalent to an ultraproduct of finite structures,and also equivalent to having the finite model property : every first order L-sentence σtrue in M has a finite model.

We start by recalling some definitions and basic results regarding ultraproducts offinite structures.

1.1 Ultraproducts of finite structures

Definition 1.1.1. If I is a set of indices and U is an ultrafilter on I, we say that I ′ ⊆ I isU-large simply if I ′ ∈ U . Suppose now that Mi : i < ω is a collection of finite structuresand U is an ultrafilter on ω.

1. We say that a property P holds for U-almost all Mi if i ∈ I :Mi |= P ∈ U .

2. We write |Mi| →U ∞ if for every N ∈ N, the set i ∈ I : |Mi| ≥ N ∈ U .

3. Given a sequence 〈ri : i ∈ I〉 of real numbers and s ∈ R, we write limn→U

rn = s to

mean: for all ǫ > 0, n ∈ ω : |rn − s| < ǫ ∈ U .

4. We denote by∏

U Mi the ultraproduct of the structures Mi : i ∈ I with respectto the ultrafilter U .

Remark 1.1.2. A trivial (but possibly clarifying) case occurs when U is a principalultrafilter of the form U = X ⊆ I : k ∈ X for some fix k ∈ I. Then we have:∏

U Mi∼= Mk, P holds for U-almost all Mn if and only if Mk |= P , |Mn| 6→U +∞ because

Mk was assumed to be finite, and for any sequence 〈rn : n < ω〉, limn→∞ rn = rk.

The fundamental theorem of ultraproducts is due to Jerzy Łoś, and provides a powerfutransfer principle between the factor structures and their ultraproduct.

9

10 LECTURE 1. BASIC CONCEPTS, EXAMPLES AND RESULTS.

Theorem 1.1.3 (Łoś, 1955). Let M =∏

U Mi be n ultraproduct of Mi : i ∈ I withrespect to an ultrafilter U on I. Then, for every first-order formula ϕ(x) = ϕ(x1, . . . , xn)and every tuple c = ([ci1]U , . . . , [cn]

iU) of elements in M , we have

M |= ϕ(c) if and only if i ∈ I :Mi |= ϕ(ci1, . . . , cin) ∈ U .

A very important property of the ultraproducts of first-order structures is the fact thatthey are ℵ1-saturated (also referred as countably saturated): for any countable A ⊆ Mand every (partial) type p(x) over M that is finitely satisfiable in M , there is a tuple cfrom M such that c realizes p(x).

Proposition 1.1.4. Let M =∏

U Mi be an ultraproduct with respect to a non-principalultrafilter U on I = ω. Then, M is ℵ1-saturated.

Proof. Suppose p(x) = ϕm(x) : m < ω is an enumeration of the formulas in p(x). Sincep(x) is finitely satisfiable in M , we have that for every k < ω the set ϕ1(M) ∩ · · ·ϕk(M)is non-empty.

By Łośtheorem, this implies that the set

S ′k := i ∈ ω :Mi |= ∃x(ϕ1(x) ∧ · · · ∧ ϕk(x))

belongs to U . Let Sk = S ′k ∩ [k,+∞). Note that these sets are U-large, Sk ⊇ Sk+1 for

every k, and⋂k<ω Sk = ∅.

Given i ∈ S1, let ki denote the largest natural number k such that i ∈ Sk, and letai ∈ ϕ1(Mi) ∩ · · ·ϕki(Mi). For each m < ω, we have by construction that

i ∈ ω : ai ∈ ϕm(Mi) ⊇ i ∈ ω : m ≤ ki ⊇ S1 ∩ Sm = Sm ∈ U .

Thus, by Łoś’ theorem, if a = [ai]U then M |= ϕm(a) for all m < ω, and so a ∈M realizesp(x).

It is sometimes useful to know the cardinality of certain ultraproducts, in order to usesome results of categoricity. When the index set is arbitrary, computing the exact size ofultraproducts is a very difficult problem even for ultraproducts of finite structures. Themain references in this subject are [21] and [25]. In the latter paper, it was shown that ifan ultraproduct of finite sets is infinite of size κ, then κℵ0 = κ.

However, when we consider ultraproducts of finite structures over a countable set ofindices, we can obtain the much simpler but still powerful result below.

Proposition 1.1.5. If M =∏

U Mi is an ultraproduct with I = ω and |Mi| →U ∞, then|M | = 2ℵ0.

Proof. Note first that

∣∣∣∣∣∏

U

Mi

∣∣∣∣∣ ≤∣∣∣∣∣∏

i∈ω

Mi

∣∣∣∣∣ ≤∣∣NN∣∣ = |R|, so |M | ≤ 2ℵ0 . For the other

inequality, given a set A ⊆ N, we can consider the function fA : N → N defined by

fA(n) =∑

k<n

χA(k) · 2k

1.1. ULTRAPRODUCTS OF FINITE STRUCTURES 11

where χA is the characteristic function of A. Consider the family F = fA : A ⊆ N.

Claim: For every n ∈ N, fA(n) < 2n and for different subsets A,B of N, n ∈ N :fA(n) = fB(n) is finite.

Proof of the Claim: It is clear that fA(n) < 2n for every fA ∈ F . Also, given twodifferent subsets A,B of N, we can even show that n ∈ N : fA(n) = fB(n) = n ∈ N :n ≤ min(AB).

Let t = min(AB), and assume without loss of generality that t ∈ A \B. Note thatfA(t+1) =

∑k≤t χA(k) ·2k = 2t+

∑k<t χA(k) ·2k = fB(t+1)+2t, so fA(t+1) > fB(t+1).

Suppose now that fA(n) = fB(n) for some n > t + 1. Then we have:

fA(n) =∑

k<t

χA(k) · 2k + 2t +∑

t<k<n

χA(k) · 2k =∑

k<t

χA(k)︸︷︷︸=χB(k)

·2k +∑

t<k<n

χB(k) · 2k = fB(n)

and substracting the first summand in both sides we obtain

2t +∑

t<k<n

χA(k) · 2k =∑

t<k<n

χB(k) · 2k

2t

(1 +

∑

t<k<n

χA(k) · 2k−t)

= 2t ·∑

t<k<n

χB(k) · 2k−t

1 +∑

t<k<n

χA(k) · 2k−t =∑

t<k<n

χB(k) · 2k−t

a contradiction, because the left hand side is an odd number while the right hand side iseven. Claim.

Consider now the set In = i ∈ I : 2n ≤ |Mi| ≤ 2n+1. The sets In are not in U , butthey form a partition of I. For each i ∈ In, let ai,j : j < 2n be a list of 2n differentelements from Mi. For every subset A ⊆ N, consider the element

aA = [ai,fA(n)]U

where n is the only natural number such that i ∈ In.Note that if A,B are different subsets of N, then

i ∈ I : aiA = aiB = i ∈ I : ai,fA(n) = ai,fB(n)= i ∈ I : i ∈ In and fA(n) = fB(n) ⊆

⋃In : f(n) = g(n)

which is a finite union of sets not in U . Thus, i ∈ I : aiA = aiB 6∈ U , and we conclude

that aA 6= aB by Łoś’ theorem. Therefore,

∣∣∣∣∣∏

U

Mi

∣∣∣∣∣ ≥ 2ℵ0.

Exercise 1.1.6.


1. If U is an ultrafilter on an index set I and for every i ∈ I, Fi is an ultrafilter on anindex set Ji, consider the index set K = (i, j) : i ∈ I, j ∈ Ji. Define the collectionW of subsets of K determined by

X ∈ W if and only if i ∈ I : j ∈ Ji : (i, j) ∈ X ∈ Fi ∈ U

Show that W is an ultrafilter on K.

2. Show that every ultraproduct of ultraproducts of finite structures is isomorphic toan ultraproduct of finite structures.

Ultraproducts of finite models have been classically used to prove results about non-definability of certain properties in classes of finite structures. Here we mention some ofthem:

Lemma 1.1.7. [Classification of ultraproducts of finite linear orders] Every infinite ul-traproduct of finite linear orders have order type of the form ω + I × Z + ω∗, where I isan ℵ1-saturated dense linear order without end points.

Proof. Let M =∏

U Li be an infinite ultraproduct of finite linear orders. First, let usconsider the formulas ϕmin(x) := ∀y(y = x ∨ x < y) and ϕmax(x) := ∀y(y = x ∨ y < x)and notice that the sentences σmin = ∃x(ϕmin(x)) and σmax := ∃x(ϕmax(x)) (asserting theexistence of minimal and maximal elements, respectively) are true in every Li. So, byŁoś’ Theorem, L has minimal and maximal elements c0, c1, respectively.

Also, notice that for every i < ω,

Li |=∀x[¬ϕmax(x)) → ∃y(x < y ∧ ∀w(x < w → (y ≤ w)))

]

That is, every element in Li different from the maximum has an immediate succesor, andsimilarly, every element in Li different from the minimum has an immediate predecessor.This property transfers to L by Łoś’ theorem and we can consider the equivalence relationon L given by:

a ∼ b⇔ there is n ∈ Z such that Sn(a) = b.

This equivalence relation is not definable, but we still have that a description of the classesa := a/ ∼ for a ∈ L given by:

a := a/ ∼=

c0, S(c0), S2(c0), . . . if a ∼ c0

c1, S−1(c1), S−2(c1), . . . , if a ∼ c1

Sn(a) : n ∈ Z if a 6∼ c0, c1

Notice that the original order in L induces a linear order in L/ ∼, given by a < b ifand only if a < b (for a 6∼ b), which has maximum elements c0 = c0/ ∼ and c1 = c1/ ∼.Let I = L/ ∼ \c0, c1. By construction, we already know that L is isomorphic toc0∪ I×Z∪ c1 ∼= ω∪ I×Z∪ω∗, so it only remains to show that I is an ℵ1-saturated denselinear order without end points, which also will have size 2ℵ0 because by Proposition 1.1.5we have

2ℵ0 = |L| = |ω + I × ω + ω| = |I|.

1.1. ULTRAPRODUCTS OF FINITE STRUCTURES 13

Suppose a, b are two elements in I with a < b, and consider the type

pa,b(x) = Sn(a) < x < S−n(b) : n < ω= ∃x1, . . . , xn, y1, . . . , yn(a < x1 < · · · < xn < x < yn < · · · < y1 < b) : n < ω

Since L is ℵ1-saturated, there is an element c ∈ L realizing pa,b(x), and we will have thata < c < b, so I is dense. A similar argument can be used to show that I has no endpoints, and this completes the proof.

Proposition 1.1.8 (Gurevich, 1988). “Having even cardinality” is not expressible in first-order logic of finite linear orders, that is, there is no a first-order sentence σ such that forevery finite structure, M |= σ if and only if M has even cardinality.

Proof. Let L = < and let K be the class of finite L-structures which are also linearorders. Suppose there is an L-sentence σ such that for all M ∈ K, M |= σ if and only if|M | is even.

Let An be the linear order (1, 2, . . . , n, <), and consider the structures M =∏

U A2n

and M ′ =∏

U A2n+1 where U is a non-principal ultrafilter on ω.By the Lemma 1.1.7, and using a back & forth argument, we can show easily that

A ∼= ω + I × Z+ ω∗ ≡ ω + I ′ × Z+ ω∗ ∼= B, and so since each A2n is even, M |= σ whichimplies M ′ |= σ, and so, A2n+1 |= σ for U-almost all n. A contradiction.

Proposition 1.1.9 (0-1 law for the finite linear orders). For every sentence σ in thelanguage L = <, there is n < ω such that either L |= σ for every finite linear order ofsize |L| ≥ n, or L |= σ for every finite linear order of size |L| ≥ m.

Proof. Suppose otherwise. Then for some sentence σ there are finite linear orders 〈Ln, L′n :

n < ω〉 such that |Ln|, |L′n| ≥ n and Ln |= σ, L′

n |= ¬σ. However, this contradicts the factthat

∏U Ln ≡∏U L

′n.

Hajek proved in [11] that “connectedness” is not first-order expressible in the class offinite graphs. Turán observed in [28] that this can be proved using ultraproducts of finitestructures.

Lemma 1.1.10. Suppose that U is a non-principal ultrafilter on ω, and 〈Gn : n < ω ∈ N〉is a collection of graphs such that every x ∈ Gn has degree 2 and Gn does not containcycles of length less than n. Then, G =

∏U Gn is a union of 2ℵ0 Z-chains.

Proof. First, consider the R-sentences

ϕ := ∀x∃y, z(y 6= z ∧R(y, x) ∧R(z, x) ∧ ∀w(R(w, x) → w = y ∨ w = z))

σk := ¬∃x1, . . . , xn(

∧

1≤i<j≤k

xi 6= xj ∧ R(x1, xk) ∧∧

1≤i<k

R(xi, xi+1)

)

asserting that the every vertex in the graph has degree exactly two, and there are nocycles of size k for any k ≥ 3. Every Gn realizes ϕ, and so G is a graph on which everyvertex has degree 2. On the other hand, for a fixed k ≥ 3, we have that

n < ω : Gn |= σk ⊇ n < ω : n ≥ k ∈ U


and by Łoś’ theorem, G |= σk for every k ≥ 3. That is, G does not contain cycles.

Using the same idea of in the proof of Lemma 1.1.7, we consider a (non-definable)equivalence relation ∼ on G given by

a ∼ b⇔ a and b are connected.

Claim: Every class Ca := a/ ∼ is isomorphic to a Z-chain.

From the claim we can conclude that |Ca| = ℵ0 and given a 6∼ b, there are no edgesbetween Ca and Cb. So, since G has cardinality 2ℵ0 , we conclude that G is isomorphic tothe disjoint union of the 2ℵ0 Z-chains given by Ca.

Proof of the Claim: Let Ca := a/ ∼ be the class of an element a ∈ G, and let b1, c1be the unique two elements in G that are adjacent to a. Put b0 = a = c0, and defineinductively the element bn+1 (resp. cn+1) to be the unique element in G such that G |=bn+1Rbn ∧ bn+1 6= bn−1 (similarly for cn+1).

Consider the function

f : Z −→ Ca

n 7−→ f(n) :=

bn if n > 0

a if n = 0

cn if n < 0

Clearly f is well-defined, because each vertex in G has degree 2. In fact, f(n)Rf(m) ifand only if |n − m| = 1. To show that f is injective, let n be the integer with minimalabsolute value such that f(m) = f(n) for some m 6= n. Without loss of generality, wemay assume that n > 0 since the case n < 0 can be proved similarly. Thus, f(n) = bn,and we have two cases:

• If m > 0, then f(m) = bm and m > n, so we would obtain the cycle

bn − bn+1 − · · · − bm = bn,

a contradiction since G does not contain cycles.

• If m < 0, then f(m) = cm and we would obtain the cycle

cm − cm−1 − · · · − c1 − c0 = a = b0 − b1 − · · · − bn = cm,

again, a contradiction because G does not contain cycles.

To show that f is surjective, if d ∈ Ca then there is some n < ω such that dist(a, d) = n.We will show by induction the following:

• If dist(a, d) = n then either d = bn or d = cn.

1.2. PSEUDOFINITE STRUCTURES AND THEORIES 15

It is clear for n = 0, 1. Suppose now that dist(a, d) = n + 1. Then, there are elementsd0, . . . , dn, dn+1 all diferent such that a = d0 − d1 − · · · − dn − dn+1 = d. By inductionhypothesis, either dn = bn or dn = cn, and also either dn−1 = bn−1 or dn−1 = cn−1.

If dn = bn and dn−1 = cn−1, then we obtain the cycle given by dn−1 = cn−1 − · · · −c1 − a− b1 − · · · − bn = dn − dn−1, a contradiction. Similarly, if dn = cn and dn−1 = bn−1.Thus, the only remaining case are dn−1 = bn−1 and dn = bn, which implies by constructiond = bn+1 ordn−1 = cn−1 and dn = cn which implies d = cn+1.

Thus, f is an isomorphism of graphs, and we conclude that Ca is a Z-chain. Claim

Proposition 1.1.11. “Being connected” is not first-order expressible in the class of finitegraphs.

Proof. Suppose σ is a sentence such that for every finite graph G, G |= σ if and only if Gis connected. For every n < ω, let An be a cycle of length 2n and let Bn be be the unionof two disjoint cycles of length n. Notice that each An is connected while each Bn is not,so An |= σ and Bn |= ¬σn. However, by the Lemma 1.1.10 M =

∏U An and M ′ =

∏U Bn

are both unions of 2ℵ0 Z-chains, so M ∼= M ′, but by Łoś’ theorem we have M |= σ andM ′ |= σ. A contradiction.

Koponen and Luosto showed in [22] using ultraproducts of finite structures that neithersimplicity nor nilpotency are first-order definable in the class of finite groups.

Let U be a non-principal ultrafilter over the set of primes P. We can show thatsimplicity is not definable in finite groups by showing that the groups G =

∏U Z/pZ and

G′ =∏

U (Z/pZ⊕ Z/pZ).

Notice that both G and G′ are infinite torsion-free divisible abelian groups of cardi-nality 2ℵ0 , so G ∼= G′ follows from the following statement:

Proposition 1.1.12. The theory of divisible abelian groups is κ-categorical for all κ ≥ ℵ1.

Proof. First, notice that if G is a divisible abelian group, we can see G as a Q-vector space

by definingp

q· x := p · z, where z is the only element satisfying the equation q · z = x, for

q > 0.

This is well-defined because if q · y1 = x = q · y2, then q · (y1 − y2) = 0 which impliesy1 = y2 because G is torsion-free. So, let G,H be two torsion-free divisible abelian groupsof the same cardinality κ ≥ ℵ1.

Any two basis of G and H as Q-vector spaces (respectively) have the same cardinalityκ, and any bijection between these basis will induce an isomorphisms G ∼= H .

1.2 Pseudofinite structures and theories

Definition 1.2.1. An L-structure M is pseudofinite if for every L-sentence σ such thatM |= σ there is a finite L-structure M0 |= σ. That is, M is pseudofinite if every sentencetrue in M has a finite model.


Definition 1.2.2. if L is a first-order language, we denote by Γf,L the common theory ofall finite L-structures.

That is, σ ∈ Γf,L if and only if σ is true in every finite L-structure.

The following result describes several equivalent definitions for a structure to be pseud-ofinite.

Proposition 1.2.3. Fix a first-order language L, and let M be an L-structure. Then thefollowing are equivalent:

1. M is pseudofinite.

2. M is elementarily equivalent to an ultraproduct of finite structures.

3. M |= Γf,L.

Proof. (2) ⇒ (3): Suppose M ≡ ∏U Mi where Mi : i ∈ I is a collection of finite

structures and U is an ultrafilter on I. Then, for every σ ∈ Γf,L we have Mi |= σ. Thus,i ∈ I : Mi |= σ = I ∈ U , and by Łoś’ theorem,

∏U Mi |= σ which implies M |= σ.

Therefore, M |= Γf,L.

(3) ⇒ (1): Let σ be an L-sentence such that M |= σ. If σ has no finite models, thenfor every finite L-structure M0 we would have M0 |= ¬σ. So, ¬σ ∈ Γf,L, and we wouldobtain M |= ¬σ, a contradiction.

(1) ⇒ (2): Suppose M is pseudofinite and let Th(M) be the collection of all L-sentences that are true in M . Let I be the collection of all finite subsets of Th(M). Forevery i = φ1, . . . , φm ∈ I, let Mi be a finite L-structure such that Mi |= φ1 ∧ · · · ∧ φm.

Let F0 be the collection of the sets of the form Xj = j ∈ I : Mj |= φ for all φ ∈ i.We will show that F0 has the finite intersection property : note that

Xi ∩Xj = k ∈ I :Mk |= φ for all φ ∈ i ∩ k ∈ I :Mk |= φ for all φ ∈ j= k ∈ I :Mk |= φ for all φ ∈ i ∪ j = Xi∪j 6= ∅.

So, F0 can be extended first to a filter F , and then to an ultrafiter (a maximal filter) U .Now we show that M ≡ ∏U Mi. If M |= σ, then the set i ∈ I :Mi |= σ ⊇ Xσ ∈ U ,

and so, by Łoś’ theorem,∏

U Mi |= σ.

Remark 1.2.4. The main difference between pseudofinite structures and ultraproducts offinite structures is that the form may omit types, while the latter are always ℵ1-saturated(if infinite). For instance, in Example 1.3.7 it is shown that the structures (Q,+) ispseudofinite, but since it is countable, it cannot be isomorphic to an ultraproduct of finitestructures.

Recall the following known result:

Theorem 1.2.5 (Keisler-Shelah isomorphism theorem). Let M,N be two L-structures.Then M ≡ N if and only if there exists an ultrafilter U (over some set of indices) suchthat

MU :=∏

U

M ∼=∏

U

N := NU .

1.3. EXAMPLES OF PSEUDOFINITE STRUCTURES AND THEORIES 17

Thus, a structure M is pseudofinite if and only if it has an ultrapower which is iso-morphic to an ultraproduct of finite structures. This gives a purely algebraic definitionof pseudofiniteness, where no sentences are mentioned.

There is a deep result in model theory due to Cherlin, Harrington and Lachlan (see[4]) that can be used to find pseudofinite models:

Theorem 1.2.6 (Cherlin, Harrington, Lachlan, 1985). All models of totally categorical(κ-categorical for each κ ≥ ℵ0) are pseudofinite.

For theories, defining pseudofiniteness might be a little bit more complicated, andthere are two versions.

Definition 1.2.7. Let T be a consistent theory, not necessarily complete.

1. T is weakly pseudofinite if whenever T |= σ then σ is true in some finite L-structure(not necessarily a model of T ).

2. T is strongly pseudofinite if whenever σ is consistent with T there is a finite structureM0 |= σ.

For instance, if T = TLO is the theory of linear orders, then T is weakly pseudofinitebut not strongly pseudofinite: the sentence ∀x, y(x < y → ∃z(x < z < y)) is consistentwith TLO but does not have finite models. However, we have the following facts (left asexercises to the reader)

Exercise 1.2.8. Show that T is weakly pseudofinite if and only if T ∪ Γf,L has somepseudofinite model.

Exercise 1.2.9. T is strongly pseudofinite if and only if T |= Γf,L if and only if everymodel of T is pseudofinite.

Exercise 1.2.10. If T is complete, then T is weakly pseudofinite if and only if T isstrongly pseudofinite.

In the light of Exercise 1.2.10, whenever T is a complete theory, we say T is pseudofiniteto mean either weakly or strongly pseudofinite.

1.3 Examples of pseudofinite structures and theories

We now will show some examples of pseudofinite structures and theories.

Example 1.3.1. For a language L, the common theory of all L-structures is weakly pseud-ofinite

Let T be the common theory of all L-structures. Note that every finite L-structure M isa pseudofinite model of T ∪ Γf,L. So, by Exercise 1.2.8, T is weakly pseudofinite.


Example 1.3.2. Let T0 be any L-theory. Let T0,f be the common first-order theory of allfinite models of T0. Then T0,f is strongly pseudofinite.Note that T0,f may not be the weakest strongly pseudofinite theory extending T0.

Of course, all these makes sense only when T0 has finite models. First, note thatT0 ⊆ T0,f by definition. Suppose now that T0,f ∪ σ is consistent, say M |= T0,f ∪ σ.Notice that if σ does not have finite models, then M0 |= ¬σ for every finite model of T0.Thus, ¬σ ∈ T0,f , but this contradicts the consistency of T0,f ∪ σ.Example 1.3.3. The theory DLO of dense linear orders is not pseudofinite.

The sentence σ := ∀x, y∃z(x < z < y)∧ “< is a linear order” does not have finite models.

Example 1.3.4. ACF :=“Theory of algebraically closed fields” is not weakly pseudofinite.

Suppose ACF ∪Γf,Lrings has pseudofinite models, say M =∏

U Mi |= ACF for a collectionMi : i ∈ I of finite rings and an ultrafilter U on I. We have two cases:

If char(M) 6= 2, consider the sentence σ1 = ∀x∃y(y2 = x) ∈ ACF . Then, for U-almostall finite fields Mi we have Mi |= σ ∧ 1 + 1 6= 0. So, the function f : Mi → Mi given byf(x) = x2 is surjective, and since Mi is finite, it is also injective.

Thus, Mi |= ∀x, y(x2 = y2 → x = y), and in particular, 1 = −1, which contradictsthat char(Mi) 6= 2.

Now, if char(M) = 2, we consider the sentence σ2 = ∀x∃y(y3 = x) ∈ ACF . Again,for U-almost all Mi we have Mi |= σ2 ∧ 1 + 1 = 0, and so the map g : Mi → Mi given byg(x) = x3 is surjective, thus injective since Mi is finite.

Therefore, Mi |= ∀x, y(x3 = y3 → x = y), and since x3− y3 = (x− y)(x2+xy+ y2) wehave that all roots of x2+xy+y2 = 0 satisfy x = y. In particular, ifMi |= ∃x(x2+x+1 = 0)then the only root is x = 1, and we haveMi |= 12+1+1 = 1+1+1 = 1 = 0, a contradictionsince Mi is a field.

Example 1.3.5. Fix a finite field F, and let T be the theory of F-vector spaces in thelanguage of additive groups. Then T is strongly pseudofinite.

The language for T is L = +, 0, (fα)α∈F and T is axiomatized by the axioms of F-vectorspaces. For instance,

(a) (V,+, 0) is an abelian group.

(b) ∀x(fα(fβ(x)) = fα·β(x)) for α, β ∈ F.

(c) ∀x, y(fα(x+ y) = fα(x) + fα(y)) for α ∈ F.

(d) ∀x(fα+β(x) = fα(x) + fβ(x)) for α, β ∈ F.

Let T ′ be the theory of infinite models of T , which is axiomatized by the axioms abovetogether with the collection of sentences

σn := ∃x1, . . . , xn

(∧

α∈F,i<j

¬(fα(xi) = xj)

): n < ω


Clearly T ′ is ω-categorical (ifM1,M2 |= T ′ are countable models, any bijection betweenbases induces an isomorphism M1

∼=M2)So, T ′ is complete, and we have two cases for a sentence σ such that T ∪ σ is

consistent: either T ′ 6|= σ, and so T ′ |= ¬σ which implies that the original model ofT ∪ σ was finite, or T ′ |= σ, and so by compactness there is some n < ω such that

(a)-(d) ∪ σn |= σ. Thus, Mn = (Fn+1,+,−→0 ) |= σ. We conclude that T is strongly

pseudofinite

Example 1.3.6. The theory of Q-vector spaces in the language of groups with a functionsymbol for scalar multiplication is pseudofinite and complete.

In this case, the theory is axiomatized by the axioms of Q-vector space, together withthe axioms

σ

rs:= ∀x

f r

s(x) + · · ·+ f r

s︸︷︷︸s-times

= fr(x)

: r, s ∈ N, s 6= 0.

This theory is ℵ1-categorical, so it is complete. To show that it is pseudofinite itis enough to find a finite model for every finite subset of the axioms. Assume thatT0 = ϕ1, . . . , ϕm ∪ σ r1

s1

, . . . , σ rnsn is a finite subset of T , where the formulas φi are

axioms for the theory of Q-vector spaces containing finitely many function symbols fα.Let N = max|ri|, |si|, height(α) : i ≤ n, fα mentioned in a formula ϕj, and pick a

prime p > N . Note that if α =r

s, we can assign f r

s(x) = r ·

(xs

)for every x ∈ Z/pZ

because gcd(p, sj) = 1. Similarly with risi

.Thus, by interpreting putting fα(x) = 0 whenever α is not mentioned in T0, we

conclude that (Z/pZ,+, 0, fα) is a finite model of T0.

Example 1.3.7. The theory Th(Q,+) is pseudofinite.

Notice that Th(Q,+) is the theory of abelian divisible groups, which is ℵ1-categorical.Let P be the collection of prime numbers, and U a non-principal ultrafilter on P. Then,∏

U Z/pZ is a divisible abelian group (as at the end of Example 1.3.6). So Th(Q,+) hasa pseudofinite model, and therefore it is pseudofinite (because it is complete).

Example 1.3.8. The theory of the random graph is pseudofinite.

Recall that the random graph is the generic Fraïssé limit of the class of finite graphs.Its theory can be axiomatized in the language L = R by the sentences

Pk,ℓ = ∀x1, . . . , xk∀y1, . . . , yℓ(∧

i,j

xi 6= yj → ∃z(∧

i,j

zRxi ∧ ¬zRyj))

We will show that each of this sentences have finite models using a probabilistic argument.Fix n ∈ N and take V = 1, 2, . . . , n. For every possible edge e ∈ [V ]2, consider the


probability space Ωe = 0e, 1e with Pe(0e) = 1 − p, Pe(1e) = p for some fixed

p ∈ [0, 1]. Let G(n, p) be the probability space Ω :=∏

e∈[V ]2

Ωe with the product measure.

Note that every element in Ω is in correspondence with a unique graphs G with set ofvertices V . So, the events in G(n, p) are simply collections of graphs on V . For example,for e ∈ [V ]2, the set Ae = x ∈ Ω : xe = 1e = G : e ∈ E(G) is the event of having e asan edge, and its probability is Pr(Ae) = Pre(1e ×

∏e′ 6=e 1 = p.

Claim 1: The events Ae are independent and occur with probability p By definition,is S = e1, . . . , ek ⊆ [V ]2, then

Pr(Ae1 ∩ Aek) = Pr

(∏

e′ 6∈S

Ωe′ ×k∏

i=1

1ei)

= 1 · pk = P (A1) · · ·P (Aek).

Consider now for k, ℓ ≥ 1, the event defined by Pk,ℓ = G ∈ G(n, p) : G |= Pk,ℓ =thecollection of graphs G such that for any disjoint U,W ⊆ G with |U | ≤ k, |W | ≤ ℓ thereis v 6∈ U ∪W such that uRv and ¬(uRw) for all u ∈ U,w ∈ W .

So, to show that the theory of the random graph is pseudofinite, it is enough to showthat given k, ℓ ≥ 1, the set Pk,ℓ(G(n, p)) 6= ∅ for some n ∈ N and some p ∈ [0, 1].

In fact, we have something stronger:

Theorem 1.3.9. For any k, ℓ ≥ 1 and every constant p ∈ (0, 1), almost every graph inG ∈ G(n, p) satisfies the property Pk,ℓ. That is,

limn→∞

Pr(Pk,ℓ(G(n, p)) = 1.

Proof. For fixed n, disjoint subsets of vertices U,W and v ∈ [n] \ (U ∪ W ), we have

Pr(∀u ∈ U, ∀w ∈ W (uRv ∧ ¬(uRw)) = p|U ||W |

(1− p)︸︷︷︸q

≥ pkqℓ. Hence,

Pr(There is no suitable v for the pair (U,W ))

= (1− p|U |q|W |)|[n]\(U∪W )| = (1− p|U |q|W |)n−|U |−|W |

≤ (1− pkqℓ)n−k−ℓ.

Notice that there are no more than nk+ℓ of pairs (U,W ) with U∩W = ∅ and |U | ≤ k, |W | ≤ℓ, since every such pair can be encoded with a function f : a1, . . . , ak ∪ b1, . . . , bℓ →1, . . . , n (if |U | < k or |W | < ℓ, the pair (U,W ) would be encoded with a non-injectivefunction). Thus, the probability that some pair U,W has no suitable element v is at most(1−pkqℓ)n−k−ℓ·nk+ℓ. So, since k+ℓ is constant and pkqℓ = pk(1−p)ℓ ≤ p(1−p) = p−p2 < 1,we conclude that

limn→∞

Pr((Pk,ℓ(G(n, p)))c) ≤ limn→∞

nk+ℓ · (1− pkqℓ)n−k−ℓ = limn→∞

nk+ℓ · rn−k−ℓ = 0

because the exponential decay dominates the polynomial growth.

In particular, given k, ℓ ≥ 1 there is some n ∈ N and some graph G with n-verticessuch that G |= Pk,ℓ. This shows that the theory of the random graph is pseudofinite.

The following is a curious note regarding the random graph:


Definition 1.3.10 (Paley graphs). Let q = pn be a prime power with q ≡ 1 (mod 4).We define the Paley graph Pq to be the graph with set of vertices V = Fq and the edgerelation defined by xRy if and only if x 6= y and (x− y) is a square.

P5

b

b

b

b

b

0

1

2

3

4

P9

b

b

bb

b

b

bb

b

P13

b

b

bbb

b

b

b

b

b bb

b

The hypothesis q ≡ 1 (mod 4) allows us to ensure that (−1) is a square in Fq, andthus R is symmetric relation. A rather technical theorem of Bollobás and Thomason (seeTheorem 10 in Ch. XIII.2 of [3]) states the following:

Theorem 1.3.11 (Bollobás - Thomason, 1981). Let U,W be disjoint sets of vertices ofPq with |U ∪W | = m, and define

v(U,W ) = v ∈ Pq \ (U ∪W ) : vRu ∧ ¬vRw for all u ∈ U,w ∈ W.

Then,

|v(U,W )− 2−mq| ≤ 1

2(m− 2 + 2m+1)q1/2 +

m

2.

Using this result as a black box, we can conclude the following:

Corollary 1.3.12. Let U be an ultrafilter on the set I = q : q is a prime power and q ≡ 1 (mod 4).Then, P =

∏U Pq is a model of the theory of the random graph.

Proof. Let U = u1, . . . , uk, W = w1, . . . , wℓ be disjoint subsets of P . Then, thesets U q = uq1, . . . , uqk, W q = wq1, . . . , wqk are disjoint subsets of Pq for U-almost all q.By Bollobás-Thomason Theorem, we have |v(U q,W q)| ≥ 1

2k+ℓq − Ck+ℓq

1/2 for some fixedconstant Ck+ℓ > 0. So, for sufficiently large q (q ≥ 2k+ℓ+1 · Ck+ℓ), we have

|v(U q,W q)| ≥ 1

2

(1

2k+ℓ

)q > 0,

and we have

q ∈ I : v(U q, wq) 6= ∅ =

q ∈ I : Pq |= ∃z

(∧

i,j

zRuqi ∧ ¬(zRwqj ))

∈ U

and by Łoś’ theorem, P |= ∃z(∧

i,j zRui ∧ ¬(zRwj)). Since this was shown for arbitrary

disjoint U,W of sizes k, ℓ respectively, we conclude that P |= Pk,ℓ for all k, ℓ ≥ 1, and soP is a model of the theory RG of the random graph.


Example 1.3.13. Almost sure theories:

Let L be a countable language and suppose K is a class of finite structures which isclosed under isomorphisms, that has only finitely many non-isomorphic models of size nfor every n ∈ N. Let µn be a uniform probability measure on the set Kn(L) = M :M is an L-structure with universe 1, . . . , n and define, for any L-sentence σ,

µ(σ) = limn→∞

µn (A ∈ Kn(L) : A |= σ) .

Definition 1.3.14. Given µ, µn,K and Kn(L) as above, we define the almost sure theoryof K as

Tas(K) = σ : σ is a first-order L-sentence and µ(σ) = 1.

Proposition 1.3.15. If M |= Tas(K) then M is pseudofinite. Moreover, σ ∈ Tas if andonly if σ is true in almost all finite L-structures.

Proof. By definition, σ ∈ Tas(K) if and only if limn→∞ µn(M ∈ Kn(L) : M |= σ) = 1,which is means by definition that σ is true in almost all finite models of K.

Now, suppose that M |= Tas(K), and let σ be an L-sentence such that M |= σ. If σdoes not have finite models, then for all n < ω and M ∈ Kn(L) we have M |= ¬σ. So,µ(¬σ) = 1, which implies ¬σ ∈ Tas. This contradicts that M |= Tas ∪ σ.

Lecture 2

Pseudofinite dimensions, forking and

simplicity.

Dimension theory is one of the most important concepts in model theory and it can beused to give a combinatorial description of the definable sets of first order structures,allowing also the use of inductive arguments to prove properties about definable sets.

One of the recurrent themes around the notions of rank is their relationship withforking-independence. It is often desired that any instance of forking (on types or formu-las) can be detected by a decrease of some dimension in the same way that any instanceof linear dependence is witnessed by a decrease in the linear dimension, or any algebraicdependence can be detected by analyzing the transcendence degree.

In [14], Hrushovski and Wagner defined the notion of quasidimension on a structureM as a way to generalize the concept of dimension allowing values in an ordered groupinstead of allowing only integer values. The main example is what I call “logarithmicpseudofinite dimension” which is defined on ultraproducts of finite structures by takingthe logarithm of the cardinality of nonstandard finite sets and factor them out by theconvex hull of the reals.

The purpose of this chapter is to study the connection between forking in a pseud-ofinite structure and the logarithmic pseudofinite dimension, and use this connections tocharacterize desirable model-theoretic properties of the ultraproduct via conditions on thepseudofinite dimension.

2.1 Measures and dimension in ultraproducts of finite

structures

Throughout this section, we will assume that L is a first order language, and C is a classof finite L-structure index by some set I, and let U be a non-principal ultrafilter on I.Suppose also that |Mi| →U ∞.

We can enrich the language L to a language L+ with 2-sorts: a sort D carrying thelanguage L and another sort OF, carrying the language of ordered rings. Also, for everyL-formula φ(x, y), add a new function symbol fφ : D|y| → OF.

23

24 LECTURE 2. PSEUDOFINITE DIMENSIONS, FORKING AND SIMPLICITY.

Given a finite structureMi ∈ C, there is a natural way to expandMi to an L+-structureKi by doing:

• D(Ki) =Mi, with its original L-structure.

• OF(Ki) = (R,+, ·, 0, 1, <).

• fφ :M|y|i → R is a function defined by fφ(b) = |φ(M |x|

i ; b)|, the cardinality of the setdefined by φ(x, b) in the structure Mi.

We consider now the ultraproduct of the structures Ki with respect to U ,

K :=∏

U

Ki =

(∏

U

Mi,R∗

).

This structure will look like a two-sorted structure, having a pseudofinite structure M inthe first sort, the non-standard reals in the second sort, and for every definable subsetX = φ(M r; b) of M a definable non-standard cardinality given by fφ(b) = |X|.

Note that since we are taking an ordered field in the second sort, we will be allowedto take sums, products, and quotients of cardinalities of definable sets, and even comparethem with rational numbers, all definably in L+.

One of the most useful features of pseudofinite structures is the fact that we can usecounting measures on the algebra of definable sets in the ultraproducts.

For a non-empty definable subsets D of M , there is a finitely-additive real valuedprobability measure µD defined as:

µD(X) := st

( |X||D|

)= lim

i→U

|X(Mi) ∩D(Mi)||D(Mi)|

.

Note that the language L+ is able to encode significant information about these mea-sures. For instance, µD(φ(x, b)) ≤ p

qif and only if M |= q · fφ(b) ≤ p · fD. The counting

measures in pseudofinite structures have been used to obtain model-theoretic proofs ofclassical results in extremal combinatorics, such as the Szemerédi’s regularity lemma, thecorrespondence Furstenberg principle, and the Szemerédi’s theorem in number theory:Every subset of the integers with positive density contains arbitrarily long arithmeticprogressions.

It is routine to show that these counting measures are finiteliy-additive real valuedprobability measures on the boolean algebra of definable subsets of M (or Mn), and byCarathéodory’s Extension Theorem it extends uniquely to a countably-additive probabil-ity measure on the σ-algebra generated by the definable sets of M .

The following result shows that, at least for definable sets, the measure µD is definable“up to infinitesimals” in the new structure K.

Proposition 2.1.1. 1. If K |= fφ(b) ≤ r · fD then µ(φ(x, b)) ≥ r.

2. If µ(φ(x, b)) < r then K |= fφ(b) ≤ r · fD.

3. If K |= fφ(b) ≥ r · fD then µD(φ(x, b) ≥ r.

2.2. THE PSEUDOFINITE DIMENSIONS 25

Proof. 1. Recall that µD(φ(x, b)) = st

( |φ(M |x|; b)||D|

)= inf

q ∈ Q : K |= fφ(b)

fD≤ q

.

So (1) is clear by definition.

2. If µ(φ(x, b)) < r, then there is q ∈ Q such that K |= fφ(b) ≤ q · fD, which clearlyimplies K |= fφ(b) ≤ r · fD.

3. If K |= fφ(b) ≥ r · fD, then for every r′ < r we have K |= ¬(fφ(b) ≤ r′ · fD). Thatis, for every r′ < r, r′ 6∈ q ∈ Q : K |= fφ(b) ≤ q · fD, and so r′ ≤ µD(φ(x, b)) forevery r′ < r. This implies that µ(φ(x; b) ≥ r.

Example 2.1.2. Notice however that there can be some “infinitesimal disagreement” be-

tween µD(φ(x; b) and fφ(b) in K: let r =p

q∈ Q ∩ (0, 1). We can define the structures

Mn in the language L = P where P is an unary predicate, given by (Mn, P (Mn)) =

(1, 2, . . . , q · n, 1, . . . , pn+ 1). Then we have that fMn

P =pn+ 1

qn=p

q+

1

n.

So, Kn |= ¬fMn

P ≤ r for every n < ω, and by Łoś’ theorem, K |= ¬(fP ≤ r). However,

µ(P (M)) = infr′ ∈ Q : K |= fP ≤ r′ = inf

r +

2

n: n ∈ N

= r.

Exercise 2.1.3. Show that this “infinitesimal disagreement” cannot occur with an irra-tional number, that is, if µ(φ(x, b)) = α 6∈ Q, then

r ∈ Q : µ(φ(x, b)) ≤ r = r ∈ Q : K |= fφ(b) ≤ r

Goldbring and Towsner developed an extension of the first order logic known as approxi-mate measure logic where they take this “infinitesimal variation” into consideration.

2.2 The pseudofinite dimensions

In this section we show that the logarithm of a non-standard finite set behaves like adimension theory, as soon as one factors out a non-trivial convex subgroup of the non-standard reals. This dimension operator will be called the pseudofinite dimension and weshow here its main features.

Definition 2.2.1. A non-empty subset S ⊆ R∗ is said to be convex if whenever s1, s2 ∈ Sand s1 < r < s2, then we also have r ∈ S.

Example 2.2.2. 1. Any non-empty interval (a, b) := x ∈ R∗ : a < x < b witha, b ∈ R∗ ∪ +∞,−∞ is a convex subset of R∗

2. Given r ∈ R∗, the monad of r defined as Sr := x ∈ R∗ : x ∈ (r − 1n, r + 1

n) for all n ∈ N

is a convex subset of R∗, but it is not an interval.

Example 2.2.3. The following are examples of convex subgroups of (R∗,+):


1. The trivial subgroup C = 0.

2. The group of infinitesimals, namely, the monad of 0 in R∗. This is the only monadwhich is also a subgroup of (R∗,+).

3. Given a non-empty subset A of R∗, we can consider the convex hull of A to be

Conv(A) =⋂

C ≤ R∗ : C is a convex subgroup and A ⊆ C..

It is clear that this is the smallest convex subgroup of R∗ that contains A. The mainexample of this kind is the subgroup Conv(Z) = Conv(R).

Proposition 2.2.4. Let α ∈ R∗, α > 0.

1. There exists a convex subgroup Cα which is the smallest convex subgroup of R∗

containing α.

2. There exists a convex subgroup C<α which is the largest convex subgroup of R∗ whichdoes not contain α.

3. There is a unique isomorphism of rings φ : Cα/C<α → R such that φ(α) = 1.

Proof. 1. Consider the group

Cα := Conv(α) =⋂

S ≤ R∗ : S is a convex subgroup of R∗ and α ∈ S.

It is clear that Cα is the smallest convex subgroup of R∗ containing α, but we writethe proof here for the sake of completeness.

Clearly α ∈ Cα since α belongs to every member in the intersection. Also, ifx, y ∈ Cα, then x, y ∈ S for every member in the intersection, and we have thatx + y,−x ∈ S for every S, which prove that S is a subgroup of (R∗,+). Finally,if s1 < r < s2 and s1, s2 are in the intersection, then s1, s2 ∈ S for each S, andsince each of the sets in the intersection is convex, r ∈ S for every S. So, Cα is alsoconvex.

2. Define C<α = x ∈ R∗ : n · |x| < α for every n ∈ N. We have that α 6∈ C<α since1 · α 6< α. Also, if s1 < r < s2 with s1, s2 ∈ Cα, we have for every n ∈ N that:

n · |r|, n · |s1 + s2|, n · | − s1| ≤ 2n ·max|s1|, |s2| < α,

which shows that C<α is a convex subgroup of R∗.

Now, suppose that C<α ( C where C is a convex subgroup of R∗. Then, there issome positive x ∈ C such that n ·x ≥ α, but in this case we have 0 < α < (n+1) ·xand since both 0 and (n + 1) · x belong to C, we conclude that α ∈ C. Thus, C<αis the largest convex subgroup of R∗ not containing α.


3. Consider the map ϕ : Cα → R given by ϕ(β) = supq ∈ Q : q ≤ βα ∈ R.

First, we leave as an exercise to show that ϕ is a ring homomorphism. Second,notice that ϕ is surjective: if r ∈ R then n < r < n + 1 for some n ∈ Z, and son · α < r · α < (n+ 1) · α. This shows that r · α ∈ Cα since it is a convex subgroupcontaining α and all its integer multiples. Now, we have ϕ(r ·α) = supq ∈ Q : q ≤r·αα

= r = r.

The kernel of the homomorphism ϕ is given by

kerϕ = x ∈ Cα : ϕ(x) = 0 =

x ∈ Cα : −1

n<x

α<

1

nfor all n ∈ N

= x ∈ Cα : n · |x| < α = C<α.

Thus, by the isomorphism theorem for rings, we have there is an isomorphism φ =ϕ : Cα/C<α → R with φ(α) = ϕ(α) = 1.

In some sense, the different convex subgroups of R∗ correspond to different “orders ofmagnitude”: if C1 ( C2 and α ∈ C1, β ∈ C2 \ C1, then α is infinitesimally smaller thanβ. This idea will play a key role in the next section when we consider the pseudofinitedimension

Note also that if C is a convex proper subgroup of R∗, then the quotient R∗/C is anabelian ordered group, with the order given by x < y in R∗/C if and only if x < y in R∗.

Definition 2.2.5. Let M =∏

U Mi be an ultraproduct of finite structures, and let C bea convex subgroup of R∗ containing Z.. For a given A ⊆ M non-empty definable subset,we define the pseudofinite dimension of A (with respect to C) as:

δC(A) = log |A|+ C,

that is, the image of log |A| under the canonical projection of R∗ onto R∗/C.

Remark 2.2.6. The hypothesis that C contains Z ensures that finite sets have dimensionzero, allowing δ to be a non-trivial dimension operator: if C does not contain Z, then Cwould be contained in the convex subgroup of the infinitesimals, and for instance, thatδ(M \ a) < δ(M) for any a ∈M , which would be absurd.

Notice that this dimension operator does not take integer values, but rather in thegroup R∗/C. The following proposition provided some evidence to consider the non-integer valued function δC as a dimension operator.

Proposition 2.2.7. Let M =∏

U Mi be an ultraproduct of finite structures, and let A,Bdefinable subsets Then:

1. If A ⊆ B, then δC(A) ≤ δC(B).

2. For any non-empty finite set X, δ(X) = 0. 1

1We can extend the dimension to the empty set by using the notation δC(∅) = −∞


3. δC(A× B) = δC(A) + δC(B).

4. δC(A ∪B) = maxδC(A), δC(B)5. If f : X → Y is a definable function, then δ(f(X)) ≤ δ(X). In particular, if X is

a definable bijection, δ(X) = δ(Y ).

6. Subadditivity: Let X, Y be definable subsets and f : X → Y be a definable surjectivefunction such that for all b ∈ Y , δC(f

−1(b)) ≤ β for some β ∈ R∗/C. Then,δ(X) ≤ δ(Y ) + β.

Proof. Assertion (1) follows directly because log is an increasing function. For (2), noticethat if X = a1, . . . , am ⊆ Mn, then δC(X) = log |X| + C = C = 0 since log |X| =logm ≤ m ∈ C. For (3), assume A =

∏U Ai and B =

∏U Bi. and notice that for every

index i we have log(|Ai ×Bi|) = log |Ai|+ log |Bi|, obtaining

δC(A×B) = log |A× B|+ C = log |A|+ log |B|+ C = δC(A) + δC(B).

For (4), suppose without loss of generality that for an U-large set of indices we have|Ai| ≥ |Bi|. Then we have |Ai ∪ Bi| ≤ 2|Ai| for U-almost all indices i, obtaining:

log |Ai| ≤ log |Ai ∪Bi| ≤ log(2 · |Ai|) = log 2 + log |Ai|log |A| ≤ log |A ∪ B| ≤ log 2 + log |A|

log |A|+ C ≤ log |A ∪ B|+ C ≤ log 2 + log |A|+ C = log |A|+ C

δC(A) ≤ δC(A ∪ B) ≤ δC(A)

because log(2 · |A|)− log |A| = log 2 ∈ C.For (5), let X =

∏U Xi and Y =

∏U(Yi). By counting in the finite structures, we

have that for U-almost all i, |f(Xi)| ≤ |Xi|, and so |f(X)| ≤ |X| which implies

δC(f(X)) = log |f(X)|+ C ≤ log |X|+ C = δC(X).

Finally, for (6), suppose that X, Y are definable subsets and f : X → Y is a definablesurjective function such that for all b ∈ Y , δC(f

−1(b)) ≤ β for some constant β ∈ R∗/C.Let r denote the element in R∗ such that β = r+C. Suppose X =

∏U Xi and Y =

∏U Yi.

Then for U-almost all i there is a definable surjective function fi : Xi → Yi, and we canchoose an element b

∗

i ∈ Yi such that |f−1i (bi)| is maximal. Then, by counting in the finite

structures, we have:

|Xi| =

∣∣∣∣∣∣

⋃

bi∈Yi

f−1i (bi)

∣∣∣∣∣∣=∑

bi∈Yi

|f−1(bi) ≤ |f−1(b∗

i )| · |Yi|.

Let b∗= [b

∗

i ]U ∈ Y . By hypothesis, since δC(f−1(b

∗)) ≤ β, there is c ∈ C such that

log |f−1(b∗)| ≤ r + c. Then we have the following:

|X| ≤ |f−1(b∗)| · |Y |

log |X| ≤ log |f−1(b∗)|+ log |Y |

log |X| ≤ (r + log |Y |+ c)

δC(X) = log |X|+ C ≤ (r + C) + (log |Y |+ C) = β + δC(Y ).


In principle, for every different convex subgroup C of R∗ there would be a differentnotion of pseudofinite dimension, and this will allow us to distinguish between variousdegrees of graininess. However, there are two main examples that will appear later in thisnotes which correspond to different special convex subgroups of R∗.

The first one is when we consider C to be the minimal interesting example: the convexhull of the standard reals, denoted Cfin. We thus denote the corresponding pseudofinitedimension by δfin.

Remark 2.2.8. In general, the map a 7→ δfin(φ(x; a)) is not definable even in the languageL+, since C = Conv(Z) and hence R∗/C are not definable.

The characteristic feature of δfin is that every possible value α for the dimensioncomes with a real-valued measure µα, defined up to a scalar multiple. Such measure ischaracterized by putting µα(X) = 0 iff δfin(X) < α, µα(X) = ∞ iff δ(X) > α, and when

δfin(X) = δfin(Y ) = α, µα(X) = st

( |X||Y |

)· µα(Y ).

In particular, we have the following:

Proposition 2.2.9. Suppose X, Y are definable sets. Then δfin(X) = δfin(Y ) if and only

if there is a natural number n such that1

n<

|X||Y | < n.

Proof. We have δfin(X) = δfin(Y ) if and only if log |X|/Conv(Z) = log |Y |/Conv(Z), if

and only if log(

|X||Y |

)log |X| − log |Y | ∈ Conv(Z). So, there is a positive integer k such

that:

−k ≤ log

( |X||Y |

)≤ k

e−k ≤ |X||Y | ≤ ek

and it suffices to pick n bigger than ek to have the desired inequality.

Corollary 2.2.10. In the language of Section 2.1, given X ⊆ Y , we have δfin(X) < δfin(Y )if and only if µY (X) = 0.

On the other hand, if we have in mind some definable set X, with α = log |X|, let C<αbe the maximal convex subgroup of R∗ not containing α, and Cα be the smallest convexsubgroup containing α. If we restrict our attention to definable sets Y with log |Y | ∈ Cα,then the corresponding dimension theory can be viewed as real valued, using the naturalisomorphisms Cα/C<α → R mapping α to 1. It can be defined directly:

δα(Y ) = st

(log |Y |α

).

A particular but important example of δα is when we consider α to be log |M |:


Definition 2.2.11. Let M =∏

U Mi be a pseudofinite structure, and A ⊆ M be a non-empty definable subset. We defined the normalized pseudofinite dimension of A δM(A),(or δC0(A) following notation from [16]) to be

st

(log |A|log |M |

)∈ [0, 1].

Alternitavely, if A =∏

U(Ai) and we put ℓi =log |Ai|log |Mi|

, (so that |Ai| = |Mi|ℓi), then

δM(A) can be also defined as δM(A) = limi→U

ℓi.

Remark 2.2.12. For the normalized pseudofinite dimension, we have some more standardproperties of the dimension. For instance, δα(M

n) = n, and more generally if |X| ≈ |M |rfor some r ∈ R, then δM(X) ≈ r.

The map δ is extended to infinitely definable sets in [15] and [16]. For ǫ ∈ R∗, chosensufficiently large and with ǫ > C, we define V0 = V0(ǫ), to be the smallest convex subgroupof R∗/C containing ǫ.

Lemma 2.2.13. Let V = V (ǫ) be the set of cuts in V0, i.e., non-empty subsets boundedabove a closed downwards. Then V (ǫ) is a semigroup, under set addition, linearly orderedby inclusion.

Proof. (V,+) is clearly a semigroup, because addition in R∗/C is associative. Now, letr, s be cuts in V . If s 6= r, we may assume without loss of generality that there is a ∈ s\ r(recall that r, s ⊆ V0).

Since r is closed downwards, it follows that x < a for all x ∈ r. Now, since S is closeddownwards, r ⊆ x ∈ V0 : x < a ⊆ S, and we conclude that r ≤ s.

The set V0 embeds into V , by the map a 7→ v : v ≤ a. We can identify V0 withits image in V under this map, and we can conclude now that any subsets of V that isbounded below has an infimum in V :

Proof. Let S ⊆ V , bounded below by a (which means that for all s ∈ S, a ⊆ s). Letα =

⋂S =

⋂s∈S s. Note that α 6= ∅, because since a is a lower bound of S, a ⊆ s for all

s ∈ S, and since S is closed downwards, v : v ≤ a ⊆ α.

• α is a cut: if b ∈ R∗/C is an upper bound for s ∈ S, it is also an upper bound forα ⊆ s So, α is bounded above.

Now, if x < y and y ∈ α, then y ∈ s for all s ∈ S, which implies x ∈ s for all s ∈ S,because all elements in S are closed downwards. Thus, x ∈ ⋂s∈S s = α.

• α = infs : s ∈ S: Assume β is a lower bound for S. Then, β ≤ s for all s ∈ S(with the order in V ), so β ⊆ s for all s ∈ S, and we have that β ⊆ ⋂

s∈S s = α,which proves that β ≤ α with the order given in V .


Definition 2.2.14. For a∧

-definable set X, define

δ(X) := infδ(D) : D ⊇ X,D definable,

the infimum evaluated in V (ǫ) for sufficiently large ǫ. Given B ⊆ M and a tuple a fromM , δ(a/B) denotes δ(tp(a/B)), and δφ(a/B) denotes δ(tpφ(a/B)), that is, the dimensionof the corresponding partial positive φ-type.

Lemma 2.2.15. Properties for the dimension, even with∧

-definable sets.

1. δ(∅) = −∞, and δ(X) = 0 for any finite definable set X.

2. If X1, X2 are∧

-definable, then δ(X1 ∪X2) = maxδ(X1), δ(X2).

3. If X1, X2 are∧

-definable, then δ(X1 ×X2) = δ(X1) + δ(X2).

4. If (αn), (βn) are decreasing sequences of cuts in V0, then

infn(αn + βn) = inf

nαn + inf

nβn.

5. If α, α′, β, β ′ ∈ V with α < α′ and β < β ′, then α + β < α′ + β ′.

6. If X =⋂nXn with X1 ⊇ X2 ⊇ · · · all

∧-definable, then δ(X) = infn δ(Xn).

7. If X is∧

-definable, f is a definable map and δ(f−1(a) ∩X) ≤ γ for some γ ∈ V0and all a, then δ(X) ≤ δ(f(X)) + γ.

Proof. 1. By convention, log 0 = log |∅| = −∞. Now, if X is finite, there is k ∈ Z suchthat |X| ≤ k, and so log |X| ≤ log k ≤ n for some n ∈ Z, so log |X| ∈ Conv(Z)which implies δ(X) = 0.

2. We have already proved this for definable sets, so we show it here for∧

-definablesets. SinceX1∪X2 contains bothX1, X2, we have that δ(X1∪X2) ≥ maxδ(X1), δ(X2).Suppose now without loss of generality that δ(X1) > δ(X2). Then, there is a defin-able set D ⊇ X2 such that δ(D) < δ(X1).

So, X1 ∪X2 ⊆ X1 ∪D, and we have

δ(X1 ∪X2) ≤ δ(X1 ∪D)

= inf δ(E) : E ⊇ X1 ∪D, E definable= infδ(E ′ ∪D) : E ′ ⊇ X1, E

′ definable (E ′ = E ∪D)

= infmaxδ(E ′), δ(D) : E ′ ⊇ X1, E′ definable

= infδ(E ′) : E ′ ⊇ X1, E′ definable (because δ(D) < δ(X1) ≤ δ(E ′))

= δ(X1).


3. We have already showed that δ(X1×X2) = δ(X1)+ δ(X2) for definable sets X1, X2.Now, if we assume X1 =

∧iX

i1 and X2 =

∧iX

i2 then we have X1×X2 =

∧i,j(X

i1×

Xj2), and so

δ(X1 ×X2) = infi,jδ(X i

1 × δ(Xj2)

= infi,jδ(X i

1) + δ(Xj2)

= infiδ(X i

1+ infjδ(Xj

2) (by (4) below)

= δ(X1) + δ(X2).

4. Since infn(αn) + infn(βn) is a lower bound for αb + βn : n < ω, we have that

infn(αn) + inf

n(βn) ≤ inf

n(αn + βn).

Suppose now the inequality is strict. Then if α = infn αn and β = infn βn we haveα+ β < infn(αn + βn), that is, infn βn = β < infn(αn + βn)− α, and so there is βmsuch that βm < inf(αn + βn)− α.

We would have then α < inf(αn + βn) − βm, and so there is some αk such thatαk < inf(αn + βn) − βm. Since (αn), (βn) are both decreasing, then if m ≥ k weconclude that αm + βm < inf(αn + βn), a contradiction.

So, infn(αn + βn) = infn(αn) + infn(βn).

2.3 Fine pseudofinite dimension and forking

Throughout the rest of this chapter, we will fork with the pseudofinite dimension δ = δfin.During this section, we will show that as well as several examples of dimension in differentmodel-theoretic contexts, the pseudofinite dimension is able to detect instances of dividingin the ultraproducts of pseudofinite structures.

This will lead to establish conditions under which the natural notion of dimension pro-vided by the pseudofinite dimension operator is equivalent to non-forking independence,or conditions that will ensure the ultraproducts are simple or supersimple.

2.3.1 Dividing and drop of pseudofinite dimension

In showing that dividing can be detected by the fine pseudofinite dimension, the followinglemma will be very important. This result roughly states that in a probability space,every infinite collection of sets with a fix positive measure have to start accumulatingwith positive measure.

Proposition 2.3.1. Let Ω be a measure space with µ(Ω) = 1, and fix 0 < ǫ ≤ 12. Let

(Ai : i < ω) be a sequence of measurable subsets of Ω such that µ(Ai) ≥ ǫ for every i < ω.

2.3. FINE PSEUDOFINITE DIMENSION AND FORKING 33

Then, for every k < ω, there are indices i1 < . . . < i2k such that

µ

2k⋂

j=1

Aij

≥ ǫ3

k

.

Proof. By induction on k. For k = 1, we have to find indices i1 < i2 such that µ(Ai1 ∩Ai2) ≥ ǫ3.

Suppose not, then for all i 6= j we have µ(Ai ∩ Aj) < ǫ3. By the truncated inclusion-exclusion, we know that for every N ∈ N,

µ

(N⋃

i=1

Ai

)≥

N∑

i=1

µ(Ai)−∑

1≤i<j≤N

µ(Ai ∩Aj)

≥ ǫ ·N −(N

2

)ǫ3

= −N2

2ǫ3 +N

(ǫ+

ǫ3

2

).

The function f(x) = −x2

2ǫ3 + x

(ǫ+ ǫ3

2

)achieve its maximum at x0 =

1ǫ2+ 1

2> 0, and if

N ∈ [x0 − 1, x0], we obtain

µ

(N⋃

i=1

Ai

)≥ f(N) ≤ f(x0 − 1)

=1

2ǫ+ǫ

2− 3

8ǫ3

≥ 1 + ǫ

(1

2− 3

8ǫ3)

(because ǫ ≤ 1/2)

> 1,

a contradiction. So, there are i1 < i2 such that µ(Ai1 ∩Ai2) ≥ ǫ1.Now, by induction hypothesis, there is a tuple (i1, . . . , ik) with i1 < . . . < i2k and

µ

2k⋂

j=1

Aij

≥ ǫ3

k

.

In fact, there are infinitey many of such tuples: otherwise, by taking ℓ to be the maximumof all indices appearing in the tuples (i1, . . . , ik), then (Aj : j ≤ ℓ + 1) would contradictthe induction hypothesis.

Let (αj : j < ω) be an enumeration of all tuples satisfying (*) and put Bj =⋂i∈αj

Ai.

By construction, µ(Bj) ≥ ǫ3k

for all j < ω. By the case k = 2, there are j1 6= j2with µ(Bj1 ∩ Bj2) ≥ (ǫ3

k

)3 = ǫ3k·3 = ǫ3

k+1

. In particular, there are at least k + 1 indicesi1 < · · · < ik < ik+1 such that

µ

(k+1⋂

j=1

aij

)≥ µ(Bj1 ∩ Bj2) ≥ ǫ3

k+1

.


Theorem 2.3.2. Let ψ(x, a) be a formula over A, and φ(x, b) a formula implying ψ(x, a)

that divides A. Then, there is an element b′ ≡A b such that δ(φ(x, b

′)) < δ(ψ(x, a).

Proof. Let D be the set defined by ψ(x, a), and suppose the result does not hold. Then

for every b′with the same type of b over A we have δ(φ(x, b

′)) = δ(D), and so there is a

natural number nb′ such that |φ(x, b′)| ≥ 1

nb′

|D|.

By compactness, there is a uniform n ∈ N such that |φ(x, b′)| ≥ 1

n|D|, since otherwise,

the L+-type given by Γ(y) = tp(b/A) ∪ |φ(x, y)| · n < |D| : n < ω would be realized in

M , and the realization b′will satisfy δ(φ(x, b

′)) < δ(D).

Now, since φ(x, b) divides over A, there is an A-indiscernible sequence (bi : i < ω) intp(b/A) such that the set φ(x, bi) : i < ω is k-inconsistent for some k < ω.

Consider now the probability measure given by µD, and let Ai := φ(x, bi) for i < ω.By the previous consideration, µD(φ(x, bi) ≥ 1

nfor every i < ω, and by Proposition 2.3.1

we have that there are indices i1 < . . . < i2k such that µ

2k⋂

j=1

Ai

≥

(1

n

)3k

> 0. In

particular, φ(x, bi1), . . . , φ(x, bik) is consistent, which is a contradiction.

The theorem above allows us to conclude that the number of possible different valuesfor pseudofinite dimensions of definable sets is a bound for the length of dividing chains,providing also a bound for the U -rank in types. We will explore this idea in Section 2.4.

We might think about two possible generalizations of Theorem 2.3.2: either changingdividing by forking or showing that the original formula (instead of replacing the param-eters by a conjugate) has lower pseudofinite dimension. The following two examples wholimitations for these attempts:

Example 2.3.3. Consider the class of finite structures Mn = ([1, 2n], En) where En is anequivalence relation with classes

[1, 2n−1, 1], [2n−1, 2n−1 + 2n−2 − 1], [2n−1 + 2n−2, 2n−1 + 2n−2 + 2n−3 − 1],

. . . , [2n−1 + 2n−2 + · · ·+ 22, 2n]

The idea here is that Mn is a set with a equivalence relations E1, E2, . . . , En with sizes

|E1| =1

2|Mn|, |E2| =

1

4|Mn|, . . . , |En| =

1

2n−1|Mn| ≥ 1.

Let M =∏

U Mn and b = [(1, 1, . . .)]U . In the ultraproduct M the relation symbolis interpreted as an equivalence relation with infinitely many infinite classes, and so theformula xEb divides over the empty set. Theorem 2.3.2 shows that there is a conjugateb′ of b over ∅ such that the formula xEb′ witnesses the drop of pseudofinite dimension.However, this drip is not witnessed by the formula xEb because

log |Mn| − log |xEn1| = log(2n)− log(2n−1) = log 2 < 1

which implies that δ(M) = δ(xEb).


Example 2.3.4. This examples is an adaptation of the classical example of the circularorder that shows that the formula x = x may fork over the empty set. Consider thestructure Mn = (Z/(3n)Z, R) where R is a ternary relation interpreted in Mn as follows:Mn |= R(b, a, c) if and only if there are integers a′, b′, c′ congruent to a, b, c (mod 3n) re-spectively, such that a′ < b′ < c′ and |c′ − a′| < n. 2 Take M =

∏U Mn, and the elements

a := [an = 0]U , b := [bn = n]U ∈M .

Claim: The formula R(x; a, b) divides over ∅.

Proof of the Claim: On each Mn consider the sequence given by

⟨(ani , b

ni ) = (n+ k · Jlog nK, n+ (k + 1) · Jlog nK) : k ≤ n

log n

⟩,

and consider in the ultraproduct the sequence given by 〈(ai, bi) = ([ani ]n∈U , [bni ]n∈U) : i < ω〉.

This is a sequence in tp(a, b/∅) which is indiscernible over the empty set, and by construc-tion we have that the set of formulas R(x; ai, bi) : i < ω is 2-inconsistent. Claim

Consider the elements in the ultraproduct M given by a1 := [an1 = 0]U , a2 := [an2 =n]U = b1 and a3 := [an3 = 2n]U = b2 and b3 = a1. Note that the formula x = x forks over∅, because it implies the disjunction

3∨

i=1

R(ai, x, bi) ∨3∨

i=1

x = ai

of formulas that divide over ∅.

a1 = b3

b1 = a2

b2 = a3

b

b

b

()

()

()

However, the set of realizations of the formula of x = x isM and it does not witness anydrop of pseudofinite dimension (δ(M) is the maximal value of the pseudofinite dimensionamong subsets of M).

Even if Theorem 2.3.2 only applies for dividing formulas, there are natural settingswhere dividing is equivalent to forking. For example forking and dividing over arbitrarysets are equivalent in simple theories , and they are also equivalent over models in theorieswith NTP2 [5].

2These structures are intended to realize the circular order in the ultraproduct.


2.3.2 Conditions on the fine pseudofinite dimension

Definition 2.3.5. The following are conditions on the fine pseudofinite dimension.

1. Attainability (Aφ). There is no sequence (pi : i ∈ ω) of finite partial positive φ-typessuch that pi ⊆ pi+1 (as sets of formulas) and δ(pi) > δ(pi+1) for each i ∈ ω. Wedenote by (A∗

φ) the corresponding (stronger) condition where the above is assumedfor all increasing sequences of finite conjunctions of (possibly negated) φ-instances.

2. Strong attainability (SA). For each partial type p(x) over a parameter set B, thereis a finite subtype p0 of p such that δ(p(x)) = δ(p0(x)).

3. Weak order definability (WODφ). There is n = nφ ∈ N such that for all a, b ∈Ms,

δ(φ(x, a)) = δ(φ(x; b)) ⇔ 1

n<

|φ(x; a)||φ(x, a)| < n.

4. Dimension Comparison in L+ (DCL+). For all formulas φ(x, y) and ψ(x, z) (witht = l(z), there is an L+ formula χφ,ψ(y, z) such that for all a ∈Ms and b ∈ M t,

χφ,ψ(a, b) ⇔ δx(φ(x; a)) ≤ δx(ψ(x, b)).

5. Dimension Comparison in L (DCL) This is as for (DCL+), except that the formulaχφ,ψ can be chosen in L.

6. Finitely many values (FMVφ) There is a finite set δ1, . . . , δk such that for eachb ∈Ms there is i ∈ 1, . . . , k with δ(φ(M r, b)) = δi.

The conditions (Aφ),(A∗φ),(WODφ) and (FMVφ) have global versions (A),(A∗),(WOD)

and (FMV), where they are assumed to hold for all φ (with k and δi in (FMV), and n in(WOD), dependent on φ).

We now proceed to present some easy observations about these conditions. Note that

(SA) ⇒ (A∗φ) for all φ⇒ (A)

We also have the following:

Lemma 2.3.6. 1. For every formula φ(x, y), the conditions ((Aφ)∧(A¬φ)∧(Aφ(x,y1)∧¬φ(x,y2)))and (A∗

φ) are equivalent.

2. The conditions (A) and (A∗) are equivalent.

Proof. This follows directly from the definitions.

Lemma 2.3.7. The conditions (WOD) and (DCL+) are equivalent.


Proof. (DCL+)⇒(WOD): This is by ω1-saturation of the ultraproduct. Namely, assume(DCL+ and that there is a formula φ(x, y) such that for all n < ω there are an, bn ∈ M |y|

with δ(φ(x; an)) = δ(φ(x; bn) but |φ(x;an)|

|φ(x,bn)|< 1

n.

Consider the set of L+-formulas given by

Ψ(y1, y2) := χφ,φ(y1, y2), χφ,φ(y2, y1) ∪ |φ(x, y1)||φ(x, y2)|

<1

n

This set is consistent and countable, so there are a, b ∈M realizing Ψ. So, |= χφ,φ(a, b) ∧χφ,φ(b, a) and δ(φ(x, a)) < δ(φ(x, b)), a contradiction.

(WOD)⇒(DCL+): Fix formulas φ(x, y), ψ(x, z), and consider the formula ρ(x, yzww1w2)given by

[(w = w1 ∧ w1 6= w2) → φ(x, y)] ∧ [(w = w2 ∧ w1 6= w2) → ψ(x, z)]

∧ [(w1 = w2 ∨ (w 6= w1, w2)) → (¬φ(x, z) ∧ ¬ψ(x, z))]

By (WOD), there is a number nρ associated with ρ. Note that φ(x, a) ≡ ρ(x; abw1w1w2)and ψ(x, b) ≡ ρ(x, abw2w1w2). Thus,

δ(φ(x, a)) ≤ δ(ψ(x, b))

⇔ There are w1 6= w2 with δ(ρ(x, abw1w1w2)) ≤ δ(ρ(x, abw2w1w2))

⇔ There are w1 6= w2 withρ(x, abw1w1w2)

ρ(x, abw2w1w2)≤ nρ

⇔M |= ∃w1, w2

(w1 6= w2 ∧ |ρ(x, abw1w1w2)| ≤ nρ · |ρ(x, abw2w1w2

)|︸︷︷︸

χφ,ψ

So, we conclude that (DCL+) holds.

Lemma 2.3.8.

1. Let f : X → Y be a definable map in M between definable sets X and Y . Supposethere is a natural number n such that for all a, b ∈ Y , 1

n|f−1(b)| ≤ |f−1(a)| ≤

n|f−1(b)| (non standard cardinalities). Then, for all a ∈ Y we have δ(X) = δ(Y ) +δ(f−1(a)).

2. Assume M satisfies (WOD). Let X be a definable set in M , f a definable map andsuppose that δ(f−1(a) = γ for all a ∈ f(X). Then, δ(X) = δ(f(X)) + γ.

Proof.

1. By counting in the finite structures, note that for every i ∈ I there are elementsa1i , a

2i ∈ Yi such that |f−1(a1i )| ≤ |f−1(b)| ≤ |f−1(a2i )| for all b ∈ Yi. On the other

hand, |X(Mi)| =∑

y∈Y (Mi)|f−1(y)| and so we have

|Y (Mi)| · |f−1(a1i )| ≤ |X(Mi)| ≤ |Y (Mi)| · |f−1(a2i )|log |Y (Mi)|+ log |f−1(a1i )| ≤ log |X(Mi)| ≤ log |Y (Mi)|+ log |f−1(a2i )|


and in the limit we obtain

log |Y |+ log |f−1(a1)| ≤ log |X| ≤ log |Y |+ log |f−1(a2)|

Now, given a ∈ Y , we have |f−1(a2)| ≤ n · |f−1(a)| and |f−1(a1)| ≥ |f−1(a)|n

. Thus,

log |Y |+ log |f−1(a)| − log n ≤ log |X| ≤ log |Y |+ log |f−1(a)|+ log n

⇒δ(Y ) + δ(f−1(a)) ≤ δ(X) ≤ δ(Y ) + δ(f−1(a)).

2. Since δ(f−1(a)) = γ for all γ and (WOD) holds, there is n = nf−1(y) such that forall a, b ∈ Y ,

1

nφ|f−1(a)| ≤ |f−1(b)| ≤ nφ · |f−1(a)|.

The result now follows from (1).

Lemma 2.3.9. Assume (Aφ) holds. Then there is m = mφ ∈ N such that there do notexists a1, . . . , am so that if pi = φ(x, aj) : j ≤ i then pi is consistent and δ(p1) > δ(p2) >· · · > δ(pm).

Proof. This follows again by compactness and ω1-saturation of M . If the result does nothold, then for every N < ω there are a1, . . . , aN such that

∣∣∣∣∣

i∧

k=1

φ(x, ak)

∣∣∣∣∣ > N ·∣∣∣∣∣

i+1∧

k=1

φ(x, ak)

∣∣∣∣∣

for each i = 1, . . . , N .

Thus, the L+-type given by p(yi : i < ω) given by

∣∣∣∣∣

i∧

k=1

φ(x, ak)

∣∣∣∣∣ > N ·∣∣∣∣∣

i+1∧

k=1

φ(x, ak)

∣∣∣∣∣ : i ≤ N,N < ω

is consistent, and by ω1-saturation of M there is a sequence 〈ai : i < ω〉 realizing thetype p. If we define pi to be

∧k≤i φ(x, ak), then we would have δ(p1) > δ(p2) > · · · ,

contradicting (Aφ).

Lemma 2.3.10. Assume (SA) holds. Then there is no sequence of definable sets 〈Sn :n < ω〉 such that Sn+1 ⊆ Sn and δ(Sn+1) < δ(Sn) for each n < ω.

Proof. Otherwise, we may consider the type p(x) := Sn(x) : n < ω. By (SA), there isa finite subtype p0 := Sn1

(x), . . . , Snk(x) with n1 < · · · < nk such that δ(p) = δ(p0) =δ(Snk) > δ(Snk+1) ≥ δ(p). A contradiction.

Lemma 2.3.11. Assume (DCL). If (FMVφ) fails for some formula φ(x, y), then T hasthe strict order property. So, in particular, T is not simple.

2.4. FORKING INDEPENDENCE AND δ-INDEPENDENCE 39

Proof. Since (FMVφ) fails, there are a1, a2, . . . such that either δ(φ(x; ai)) > δ(φ(x; ai))for all i < ω, or δ(φ(x; ai)) < δ(φ(x; ai)) for all i < ω. Then, there is a definable pre-orderwith infinite chains given by

y ≤ y′ ⇔ δ(φ(x; y)) ≤ δ(φ(x, y′)) ⇔ χφ,φ(y, y′).

Lemma 2.3.12.

1. Assume Th(M) has the SOP, then (FMV) fails.

2. If M has (DCL) and Th(M) has elimination of imaginaries, then (FMV) holds andso Th(M) does not have the SOP.

Proof. The idea here is that from an order with infinite chains we can get a dense linearorder. From there, we can use ω1-saturation: from a0 < aǫ < a1 with δ([a0, aǫ]) <δ([a0, a1]), we can iterate this argument to obtain δ([a0, a1]) > δ([a0, aǫ1]) > δ([a0, aǫ2]) >· · · , obtaining an infinite drop of dimensions.

For a more detailed proof: For (1), let ψ(u, v) be a formula defining a preorder ≺ inM t with an infinite chain. By Erdös-Rado and ω1-saturation there is an L+-indiscerniblesequence in M t with ai ≺ aj if and only if i < j.

Let χ(x, uv) be express u ≺ x ≺ v. Hence, since χ(a0, an) ⊇n⋃

i=0

χ(ai, ai+1) and

fχ(a0, a1) = fχ(ai, ai+1), we have fχ(a0an) ≥ n · fχ(a0a1), for every n > 0.So, by L+-indiscernibility, fχ(a0a2) ≥ n ·fχ(a0a1) for each n, and so, δ(χ(M t, a0a2)) >

δ(χ(M t, a0a1)). It follows by compactness that the set δ(χ(M t; a0ai)) : i > 0 is infinite.

For (2), let us assume (DCL) and (EI), and suppose for a contradiction that δ(φ(M r; a)) :a ∈ Ms is infinite. Let ψ(u, v) express δ(φ(M r, u)) ≤ δ(φ(M r, v)). Then, ψ defines apreorder on Ms.

Let E be the equivalence relation defined by putting E(u, v) ⇔ ψ(u, v) ∧ ψ(v, u).Then ψ induce on Ms/E an empty-definable total order ≺. By EI, for some t there is anempty-definable function g :Ms →M t with E(u, v) ⇔ g(u) = g(v).

There is then an empty-definable total order ≺ on I = g(Ms) given by a ≺ b ⇔g−1(a) < g−1(b).

Since I is pseudofinite, we may find (by ω1-saturation) a sequence of subintervalsI ⊇ J0 ⊇ J1 ⊇ J2 ⊇ · · · with |Ji| = 2|Ji+1| (non-standard cardinalities). Since theintervals are uniformly definable, this contradicts (WOD) and hence (DCL).

Example 2.3.13. It is easy to produce examples with (A) or (SA) but without (FMV).For instance, we can consider the Example 2.3.3.

2.4 Forking independence and δ-independence

Definition 2.4.1. Let a be a tuple and A,B be countable subsets of M . We say that ais δ-independent of B over A (denoted by a |

δ

AB) if δ(a/AB) = δ(a/A).


Remark 2.4.2. With a, A,B as in the previous definition, then a 6 |δ

AB there is a formula

θ(x) ∈ tp(a/AB) such that for all ψ ∈ tp(a/A), δ(θ(x)) < δ(ψ(x)).

If a 6 |δ

AB, then we would have

δ(a/AB) = inf δ(φ(x)) : φ(x) ∈ tp(a/AB)< δ(a/A) = infδ(ψ) : ψ ∈ tp(a/A)

then there is θ(x) ∈ tp(a/AB) such that δ(θ(x)) < δ(a/A), and so, δ(θ(x)) < δ(ψ(x)) forall ψ(x) ∈ tp(a/A).

Remark 2.4.3. Note that in the proof of case k = 1 in Proposition 2.3.1, we actuallyfound a number N = N(ǫ) such that if A1, . . . , AN have measure at least ǫ, there are1 ≤ i < j ≤ N such that µ(Ai ∩Aj) ≥ ǫ3.

We are interested in the properties of δ-independence, and we want to see until whichextent the δ-independence satisfies standard properties of the non-forking independencein simple theories.

Lemma 2.4.4 (Additivity for δ). Assume (DCL) and (FMV), and let A be a countableset of parameters from M |= T . Let a ∈M r, b ∈Ms, then δ(ab/A) = δ(a/Ab) + δ(b/A).

Proof. Since A is countable, we may assume without loss of generality A = ∅. Let(φn(y) : n < ω) enumerate the formulas in tp(b) and (ψn(x, b) : n < ω) enumeratetp(a/b). We may suppose that ψn+1 → ψn and φn+1 → φn for each n < ω.

Let P be the set of realizations of tp(ab) in M (P ⊆M r×Ms), and put ǫn := δ(φn(y)),γn := δ(ψn(x, b)). By (DCL) and (FMV), for each n there is a formula ρn(y) ∈ Lexpressing that δ(ψn(x, y)) = γn, as follows:

Let χ(y1, y2) be an L-formula such that M |= χ(b1, b2) ⇔ δ(ψn(x, b1)) ≤ δ(ψn(x, b2)).

By (FMV), there are finitely many values for the set δ(ψn(x, b′)) : b

′ ∈ Ms (say k

values), so if γn is the j-th of these values, then δ(ψn(x, b′)) = γn if and only if

M |= ∃y1, . . . , yj−1, yj+1, . . . , yk

∧

h<i∈1,...,k−j

(χ(yh, yi) ∧ ¬χ(yi, yh))

∧ (χ(yj−1, b′) ∧ ¬χ(yj−1, b

′)) ∧ (¬χ(yj+1, b

′) ∧ χ(b′, yj+1))

The last formula is ρn(y).Since ρn(y) ∈ tp(b), there is a formula φmn such that φmn ⊢ ρn. By refining the

sequence, we can suppose that φn ⊢ ρn. Let Pn be the set defined by φn(y) ∧ ψn(x, y).Claim: δ(Pn) = ǫn + γn for each n.

Proof of the claim: Given b′

n |= φ(y), since δ(ψn(x, b′)) = δ(ψn(x, b)), there exists an

integer N ∈ N (uniform, by compactness and saturation) such that

1

N|ψn(x, b)| ≤ |ψn(x, b

′)| ≤ N · |ψn(x, b)|


By counting in the finite structures Mi, there are b1

i , b2

i ∈ φn(Mi) such that

|ψn(M ri , b

1

i )| · |φn(Mi)| ≤ Pn(Mi) ≤ |ψn(M ri , b

2

i )| · |φn(Mi)|.

So, for U-almost all i, we have

1

N|ψn(M r

i , bi)| · |φn(Mi)| ≤ |Pn(Mi)| ≤ N · |ψn(M ri , bi)| · |φn(Mi)|

log1

N+ log |ψn(M r

i , bi)|+ log |φn(Mi)| ≤ log |Pn(Mn)| ≤ log1

N+ log |ψn(M r

i , bi)|+ log |φn(Mi)|

By taking limits and quotient by Conv(Z) we obtain

ǫn + γn = δ(ψn(x, b)) + δ(φn(y)) ≤ δ(Pn) ≤ δ(ψn(x, b)) + δ(φn(y)) = ǫn + γn. X

Note that P =⋂n<ω Pn, so

δ(tp(ab/A)) = δ(P ) = infn(δ(Pn)) = inf

n(ǫn + γn)

= infn(ǫn) + inf

n(γn) = δ(tp(b)) + δ(tp(a/Ab)).

Proposition 2.4.5. The following are properties for the δ-independence.

1. Existence: Given countable sets A ⊆ B and p ∈ Sr(A) (for any r ∈ N), there isa |= p such that a |

δ

AB.

2. Monotonicity and Transitivity: if A ⊆ D ⊆ B, then

aδ

|A

B ⇔(a

δ

|A

D and aδ

|D

B

)

3. Finite character: If a 6 |δ

AB then there is a finite subset b ⊆ B such that a 6 |

δ

Ab.

Proof.

2. If δ(a/AB) = δ(a/A), we have δ(a/AB) ≤ δ(a/D) ≤ δ(a/A) = δ(a/AB), and so,δ(a/AD)δ(a/A). Similarly, δ(a/AB) ≤ δ(a/BD) ≤ δ(a/AD) ≤ δ(a/A) ≤ δ(a/AB),and so δ(a/BD) = δ(a/AD) = δ(a/A). Thus, a |

δ

AD and a |

δ

DB.

The converse follows from δ(a/AB) = δ(a/AD) = δ(a/A).

1. Given a partial type q over B and a formula φ(x, b) over B, note that if δ(q) = δ0then either δ(q ∪ φ(x, b)) = δ0 or δ(q ∪ ¬φ(x, b)) = δ0. Otherwise, there wouldexists a formula ψ ∈ q such that δ(ψ ∧ ¬φ) < δ0 and δ(ψ ∧ φ) < δ0 and we wouldhave δ(q) = δ0 ≤ δ(ψ) = δ((ψ ∧ φ) ∪ (ψ ∧ ¬φ)) = maxδ(ψ ∧ φ, δ(ψ ∧ ¬φ) < δ0, acontradiction.


3. Suppose a 6 |δ

AB, then δ(a/AB) < δ(a/A), so there is a formula φ(x, b) over B such

that δ(tp(a/A) ∪ φ(x, b)) < δ(a/A). Then, a 6 |δ

Ab.

Proposition 2.4.6. Under further assumptions, we have

4. Local character: (uses (A)) For every a and B ⊆M there is a countable set A ⊆ Bsuch that a |

δ

AB.

5. Invariance: (uses (DCL)) If α ∈ Aut(M), then a |δ

AB if and only if α(a) |

δ

α(A)α(B).

6. Symmetry: (uses (DCL) and (FMV)) a |δ

Ab if and only if b |

δ

Aa.

7. Algebraic closure: (uses (DCL)) If A ( B ⊆ acleq(A), then δ(a/A) = δ(a/B).(whereδ is defined in the natural way for formulas with parameters in Meq)

Proof. 4. Let p := tp(a/B). By (A), for each formula φ(x, y) ∈ L there is a φ-formula ψφ(x, bφ) (collection of φ-instances) such that δφ(a/B) := δ(tpφ(a/B)) =

δ(ψφ(x, bφ)).

Let A be the collection of all elements in tuples bφ, when φ varies. Then, |A| ≤ ℵ0

and a |δ

AB.

5. Suppose a |δ

AB and α ∈ Aut(M). Note that for every formula φ(x, b) ∈ tp(a/AB)

there is ψ(x, c) ∈ tp(a/A) such that δ(ψ(x, c)) ≤ δ(φ(x, b)).

By (DCL), there is an L-formula χψ,φ such that M |= χψ.φ(b, c), and since it is in-variant under automorphisms of M , we have M |= χψ,φ(c, b), hence, δ(ψ(x, α(c))) ≤δ(φ(x, α(b))), and we conclude that α(a) |

δ

α(A)α(B).

6. It suffices to show that a |δ

Ab implies b |

δ

Aa. By additivity of δ (which uses (FMV)

and (DCL)) we have

δ(a/A) + δ(b/Aa) = δ(ab/A)

= δ(b/A) + δ(a/Ab)

= δ(b/A) + δ(a/A) (because aδ

|A

b)

So, δ(b/Aa) = δ(b/A), which implies b |δ

Aa.

7. Suppose ψ(x, b) is an algebraic formula such that ψ(a, b) holds, where b is possiblyan imaginary tuple in acleq(A), and let b = b1, b2, . . . , bk be the conjugates of b overA. Then, δ(ψ(x, bi)) = δ(ψ(x, b)) for each i = 1, . . . , k by (DCL), and so there is aformula ρ(x) ∈ tp(a/A) such that ρ(x) ≡ ∨n

i=1 ψ(x, bi), and thus,

δ(ρ(x)) = maxi

δ(ψ(x, bi)) = δ(ψ(x, b)).


Remark 2.4.7. If we assume (SA), then for local character we have that the subsetA ⊆ B can be taken to be finite: there is a single formula φ(x, b) ∈ tp(a/B) such thatδ(a/B) = δ(φ(x, b)), and we can take A to be the tuple b.

2.4.1 Simplicity and forking

Definition 2.4.8. Let T be a complete theory and M an ω1-saturated model of T , fromwhich the parameters will be taken.

1. A formula φ(x, y) has the tree property (with respect to T ) if there is k < ω and asequence (aµ : µ ∈ ω<ω) such that:

(a) For every µ ∈ ω<ω, the set φ(x, aµi : i < ω is k-inconsistent.

(b) For every σ ∈ ωω, the set φ(x, aσi : i ∈ ω is consistent

2. The theory T is simple if no formula φ has the tree property with respect to T .

3. A dividing chain of length α for φ is a sequence (ai : i < α) such that⋃

i<α

φ(x, ai) is

consistent and φ(x, ai) divides over aj : j < i for all i < α.

4. A simple theory T is low if for every formula φ there is nφ < ω such that there isno dividing chain of length nφ for φ.

Lemma 2.4.9. Let D be an A-definable subset of M r in the language L, and let φ(x, y)be an L-formula with |x| = r and |y| = s. Let (ai : i ∈ I) be an L+-indiscernible sequenceover A of elements of Ms.

Put Di := φ(M r; ai) for each i ∈ I, and suppose Di ⊆ D and (Di : i ∈ I) isinconsistent. Then, there is some i ∈ I such that δ(Di) < δ(D).

Proof. Suppose otherwise. By L+-indiscernibility of the sequence (ai : i ∈ I), there issome n ∈ N such that |Di| ≥ 1

n|D|. The rest of the proof follows as in the proof of

Theorem 2.3.2.

Theorem 2.4.10. 1. Assume (A) holds. then T is simple and low.

2. If (A) and (DCL) hold, then (FMV) holds.

Proof. For (1), it is showed in ([29],Proposition 2.8.6) that φ(x, y) has the tree propertythen φ has a dividing chain of arbitrary length. We will show that for every L-formulaφ(x, y) there is m := mφ < ω such that φ does not have dividing chains of length m.

Suppose for a contradiction that there is a sequence (aj : 1 ≤ j ≤ m + 1) such thateach φ(x, aj) divides over ai : i < j. Since (A) holds, by Lemma 2.3.9 to obtain acontradiction it will suffice to show that there is a sequence (bj : 1 ≤ j ≤ m + 1) suchthat tpL(aj : 1 ≤ j ≤ m+ 1) = tpL(bj : 1 ≤ j ≤ m+ 1) and

δ(φ(x, b1) ∧ · · · ∧ φ(x, bk) ∧ φ(x, bk+1)) < δ(φ(x, b1) ∧ · · · ∧ φ(x, bk)).


We construct the sequence (bj : 1 ≤ j ≤ m + 1) by induction, starting with b1 = a1.Suppose now that b1, . . . , bk have been constructed. As tpL(b1, . . . , bk) = tpL(a1, . . . , ak),there is c ∈M such that tp(b1, . . . , bk, c) = tpL(a1, . . . , ak, ak+1). Put now A = bi : i ≤ kand D = φ(x, b1 ∧ · · · ∧ φ(x, bk), and let di : i < ω be an A-indiscernible sequence wit-nessing the dividing of the formula φ(x, c) over A. Using Erdös-Rado, compactness andω1-saturation of M , we may assume that (di : i < ω) is A-indiscernible in the languageL+. Put now, Di = D ∧ φ(x, di). By Lemma 2.4.9 there is i < ω such that δ(Di) < δ(D).Put bk+1 = di.

For (2), notice that by Lemma 2.3.11, if (DCL) holds and (FMVφ) fails, T is notsimple. However, this contradicts (1).

Theorem 2.4.11. Let A,B be countable subsets of M , and a a tuple from M .

1. Assume (A) and (DCL) hold, then a |AB ⇒ a |

δ

AB.

2. Assume (SA) and (DCL). Then a |δ

AB ⇒ aAB.

In particular, (SA) and (DCL) imply that δ-independence is equivalent to non-forkingindependence.

Proof. Recall that any independence relation satisfying existence, monotonicity, transitiv-ity, finite character, local character, invariance, symmetry and algebraic closure, is impliedby forking-independence, as it was proved in [19].

Note that all of them hold by Propositions 2.4.5, 2.4.6 and the fact that (A) and (DCL)implies (FMV).

However, we show the proof of (1) here: Suppose a 6 |δ

AB. Hence δ(a/AB) < δ(a/A),

and so there is a formula φ(x, b) ∈ tp(a/AB) such that δ(φ(x, b)) < δ(a/A).

We must show that a 6 |AB, but it suffices to show that a 6 |A

b, i.e., that tp(a/Ab)forks over A.

Suppose for a contradiction that a 6 |Ab. By Existence for δ, and ω1-saturation, there

is a sequence (bi : i < ω) in M such that bi |= tp(b/A) and bi |δ

Abj : j < i for all i

(constructed by induction, of course).

By ω1-saturation and Erdös-Rado, we may even assume that (bi : i < ω) is A-indiscernible in L+.

Let p := p(x, b) := tp(a/Ab). Since a |Ab, the set φ(x, bi) : i ∈ ω1 is consistent,

realized by a′ say. By (DCL) and (FMV), δ(φ(x, bi)) = δ(φ(x, b)) and δ(a/A) = δ(a′/A).Thus,

aδ

6 |A

bi for all i. (†)

On the other hand, by local character there is i ∈ ω1 such that a′ |δ

Ab<ibj : j ∈ ω1.

In particular, a′ |δ

Ab<ibi. However, using symmetry and transitivity of |

δ, we obtain a


contradiction as follows:

a′δ

|Ab<i

bi ⇒ biδ

|Ab<i

a′ and biδ

|A

b<i (by symmetry, and construction)

⇒ biδ

|A

a′ (by transitivity for δ)

⇒ a′δ

|A

bi (by symmetry)

contradicting (†).For (2), suppose a 6 |A

B. We must show that δ(a/AB) < δ(a/A). By (SA), thereis a formula ψ(x) ∈ tp(a/A) such that δ(ψ(x)) = δ(tp(a/A)). Since a |A

B, there is a

formula φ(x, b) ∈ tp(a/AB) that forks over A, and we may suppose φ(M r; b) ⊆ ψ(M r).By (A), T is simple, and so dividing and forking agree. Hence, by Theorem ??,

there is some b′ |= tp(b/A) such that δ(φ(x, b

′)) < δ(ψ(x)). Now, by (DCL), if α is an

automorphism of M fixing A such that α(b) = α(b′), then φ(α(a, b

′)) holds, which implies

α(a) 6 |δ

Ab′, and by invariance, a 6 |

δ

Ab.

Remark 2.4.12.

(a) The above proof shows, assuming (SA) and (DCL), that a formula φ(x, b) forksover A if and only if for every L(A)-formula ψ(x) that is consistent with φ we haveδ(φ ∧ ψ) < δ(ψ). The direction (⇐) only requires (A) and (DCL).

(b) Also, just under (SA), the proof of (2) yields the following: Suppose a 6 |AB. Then

there is a′B′ such that tpL(aB/A) = tpL(a′B′/A) and a′ 6 |

δ

AB′.

Note here that δ-dimension is not part of the L-type, and is not preserved in generalunder automorphisms of the ultraproduct (unless (DCL holds, of course).

Theorem 2.4.13. Assume (SA). Then T is supersimple.

Proof. Suppose for a contradiction that T is not supersimple. Then there are countablesets B0 ⊆ B1 ⊆ · · · and a type p over B =

⋃i<ω Bi such that for all i < ω, p Bi+1

forksover Bi. Let a be a realization of p.

Claim: For every n < ω, we can build sets B′0 ⊆ B′

1 ⊆ · · · ⊆ B′n along with tuples an

such that tp(anBn′) = tp(aBn) and δ(an/B

′i+1) < δ(an/B

′i) for every i < n.

Proof of the Claim: By induction, suppose these tuples and sets have been found for afixed n < ω. Notice that there is a set B∗

n such that δ(an/B′0 . . . B

′nB

∗n) = δ(a/B0 . . . BnBn+1).

Thus, an 6 |B′≤n

B∗n, so by Remark 2.4.12 there is an+1B

′n+1 such that

tp(an+1B′n+1/B

′0 . . . B

′n) = tp(aBn+1/B0 . . . Bn)

and δ(an+1/B′n+1) < δ(an+1/B

′n). Claim

Let B′ :=⋃n<ωB

′n, pn := tp(an/B

′n) and p′ =

⋃n<ω pn. Then p′ ∈ S(B′), and

δ(p′ Bn+1) < δ(p′ B′

n) for each n, contradicting Lemma 2.3.10


2.5 Fine pseudofinite dimension and stability

In the following proposition, we characterize among ultraproducts M satisfying (A) whenM is stable (and NIP). This characterization if made locally, at the level of formulasφ(x, y).

Theorem 2.5.1. Assume (A∗φ) holds. Then the following are equivalent:

1. φ(x, y) has the independence property

2. φ(x, y) is unstable.

3. For some d there is a d-definable sets D ⊆ M r and a sequence (ai : i ∈ ω) L+-indiscernible over d such that

δ(D) = δ

(D ∧

∧

i∈ω

φ(x, ai)

)

and µD(φ(x, ai) ∧ φ(x, aj)) < µD(φ(x, ai)) for all i < j.

Proof. (1) ⇒ (2) is clear: if φ(x, y) has the independence property witnessed by the se-quences (ai : i < ω), (bW : W ⊆ ω) then we have φ(ai, bW ) holds if and only if i ∈ W . By

taking b′

j := b1,...,j, we obtain φ(ai, b′

j) holds if and only if i ≤ j. Thus, φ(x, y) is unstable.

(2) ⇒ (3) : If φ(x, y) is unstable, there are sequences (bi : i < ω) and (aj : j < ω) suchthat M |= φ(bi, aj) if and only if i > j. By (A∗

φ), there is a number mφ such that there arenot finite partial φ-types D1 ⊇ D2 ⊇ · · · ⊇ Dmφ such that δ(D1) > δ(D2) > · · · > Dmφ .

So, we can find a set D, defined by a partial φ-type such that

• bi : i < ω ⊆ D.

• If bi : i ∈ ω ⊆ D′ and D′ is a finite partial φ-type, then δ(D′) ≥ δ(D).

Suppose (2) is false. Let d be the parameters needed to define D and take (aibi : i ∈ω + 1) an L+-indiscernible sequence over d, with bi ∈ D for all i ∈ ω + 1 and such thatM |= φ(bi, aj) if and only if i > j.

From L+-indiscernibility, it follows that µD(φ(x, ai)) is constant. Also, by the mini-mality in the choice of D, we have that δ(D) = δ(D ∧ φ(x, a0)), and so there is a natural

number M such that |D∩φ(x, a0)| ≥ |D|M

. Again, by L+-indiscernibility, |D∩φ(x, ai)| ≥ |D|M

for each i < ω.Since (2) is false, and by L+-indiscernibility, we must have µD(φ(x, ai) ∧ φ(x, aj)) =

µD(φ(x, ai) for every i < j < ω+1, which implies µD(φ(x, ai)∧¬φ(x, aj)) = 0, and so wehave δ(D ∧ φ(x, ai)) > δ(D ∧ φ(x, ai) ∧ ¬φ(x, aj)) whenever i < j < ω + 1.

Finally, put D′ = D ∩ φ(M r, a0) ∩ ¬φ(M r, aω). Notice that D′ is defined by a finitepartial φ-type, bi : i < ω ∈ D′ and δ(D′) < δ(D). This contradicts the minimality ofD.

(3) ⇒ (1) :Let D be a d-definable set, (ai : i ∈ ω) be an L+-indiscernible sequenceover d with φ(x, ai) ⊆ D and such that

2.5. FINE PSEUDOFINITE DIMENSION AND STABILITY 47

• δ

(D ∧

∧

i<ω

φ(x, ai)

)= δ(D).

• µD(φ(x, ai) ∧ φ(x, aj)) < µD(φ(x, ai) for all i < j.

Put Di := D ∧ (φ(x, a2i) ∧ ¬φ(x, a2i+1)) for i < ω. Since µD(φ(x, a2i) ∧ φ(x, a2i+1)) <µD(φ(x, a2i)), we have µ(Di) > 0. Moreover, since the sequence (ai : i < ω) is L+-indiscernible over d and D is d-definable, we may assume that µ(Di) = µ for someconstant real µ > 0.

By Proposition 2.3.1 and L+-indiscernibility, we conclude that µ(D1∩· · ·∩Dk) > 0 forany k < ω. In particular, this shows that the alternation number of the formula φ(x, y)in the sequence (ai : i < ω) is infinite, hence φ(x, y) has the independence property.

Example 2.5.2 (Supersimplicity does not imply (SA)). Let L = Pi : i < ω be thelanguage consisting of countably many unary predicates. For each n < ω, let Mn bea finite structure with domain [nn] := 1, . . . , nn such that Pi(Mn) ⊇ Pi+1(M

n) and|Pi(Mn)| = nn−i for each 1 ≤ i ≤ n, and Pi(Mn) = ∅ for i > n.

Notice that for every a ∈ M and A ⊆ M , tp(a/A) is completely characterized by theset i ∈ ω : M |= Pi(a). Thus, we have elimination of quantifiers, and this also showsthat the only dividing formulas are the algebraic ones (defined by disjunction of formulasof the form x = a). So, Th(M) is superstable of U -rank 1.

However, (SA) does not hold since δ(Pi(M)) > δ(Pi+1(M)) for all i < ω.

Example 2.5.3 (Superstability does not imply Aφ). For each n < ω, let (Mn, E) be thestructure where Mn has i = 1nni elements and E is interpreted as an equivalence relationwith a class of size ni for each 1 ≤ i ≤ n. Let an,i ∈ Mn be an element in the class of sizenn−i.

Notice that the theory Th(M) is simply the theory of an equivalence relation withinfinitely many infinite classes, thus Th(M) is superstable of U -rank 2.

Consider the formula φ(x, y) := ¬(xEy). We will show that attainability fails for theformula φ. Consider the elements (bi := [an,i]n∈U : i ∈ ω), and consider the positive φ-typep := φ(x, bi) : i < ω = ¬(xEbi) : i < ω.

Claim: For each t < ω, δ(xEbt) > δ(p).

Proof of the Claim: Otherwise, δ(xEbt) ≤ δ(p) = infk<ω

δ

(∧

i≤k

¬(xEbk))

. In particular, this

implies that

δ(xEbt) ≤ δ

(∧

i≤t+1

¬(xEbi)).


So, there is a constant N ∈ N such that, for U-almost all n, we have

log |xEan,t| − log

∣∣∣∣∣∧

i≤t+1

¬(xEan,i)∣∣∣∣∣ ≤ N

log(nn−t)− log

(∑

t+2≤i≤n

nn−i

)≤ N

log(nn−t)− log(n · nn−(t+2)) ≤ N

(n− t) log n− logn− (n− (t + 2)) logn ≤ N

(log n) · (n− t− 1− n+ t+ 2) = log n ≤ N

which is clearly false for sufficiently large n. Claim.

From the Claim, we conclude that given finitely many bi1 , . . . , bik , the set defined byk∧

j=1

φ(x, bij ) contains xEbt for t > i1, . . . , ik and we have

δ

(k∧

j=1

φ(x, bij )

)≥ δ(xEbt) > δ(p)

Thus, (Aφ) does not hold.

Lecture 3

Applications of pseudofinite structures

to combinatorics

3.1 Introduction

The ultraproducts of finite structures can be used to provide a bridge between infinite andfinite structures, and have been used to establish some results in asymptotic combinatorics(or extremal combinatorics, as it is most commonly refer), which is a branch of combi-natorics that studies the properties of classes of finite structures that contain arbitrarilylarge finite structures, and in particular those properties that hold “in the limit” of theclasses.

The use of ultraproducts in this context have produced striking results whose proofsrely on tools from “infinitary” mathematics. As examples of this results we can mentionnew proofs of Szemerédi’s Regularity Lemma in graph theory and Szemerédi’s Theoremin number theory (Goldbring and Towsner, [10]), a striking progress in the so-calledFreiman’s problem in multiplicative combinatorics (Hrushovski, [15]), and an algebraicregularity lemma that yield to a classification of expandes in finite fields (Tao, [26]).

Roughly speking, we can describe the general strategy as follows: Suppose C is a classof finite structures (e.g. graphs, groups, fields, etc.) and we would like to show thatcertain property P holds for every sufficiently large structure in C. For a contradiction,assume otherwise. Then, there are structures 〈Mn : n < ω〉 in C such that:

• limn→∞

|Mn| = ∞.

• Mn does not satisfy the property P for each n ∈ ω.

This is usually referred to as a sequence of counterexamples of the property P. Fromthat we can take the ultraproducts M =

∏U Mn of the structures Mn with respect to a

non-principal ultrafilter U .

If it is possible to give a first-order description of the property P (possibly in anextended language), we would have by Łoś’ theorem we also have that M does not satisfy

49

50LECTURE 3. APPLICATIONS OF PSEUDOFINITE STRUCTURES TO COMBINATORICS

P 1. On the other hand, if it is possible using methods from infinitary mathematics (e.g.measure theory, functional analysis, comparison finite vs. infinite cardinalities, etc.) thatthe infinite structure M must satisfy the property P.

To illustrate this method a little bit further, we give a proof of the celebrated Ramsey’sTheorem.

Theorem 3.1.1 (Ramsey, 1930). For every k ∈ N there is a number R(k) ∈ N such thatevery graph G with |V (G)| ≥ R(k) contains either a clique or an anticlique of size k.2

Proof. If the statement of the theorem does not hold, then for a fix k ∈ N and for everyn < ω there is a graph Gn such that |V (Gn)| ≥ n and Gn does not contain neither cliquesnor anticliques of size k.

In this particular case, the property P we want to show can be easily stated in thelanguage of graphs L = R:

P := ∃x1, . . . , xk(

∧

1≤i<j≤k

xi 6= xj ∧(

∧

1≤i<j≤k

xiRxj ∨∧

1≤i<j≤k

¬(xiRxj)))

Let U be a non-principal ultrafilter over N, and take G =∏

U Gn. Since |Gn| ≥ n forevery n and 〈Gn : n < ω〉 was a sequence of counterexamples, we know that G is infiniteand by Łoś’ theorem, G |= ¬P.

The tool from infinitary mathematics we will use in this case is simply the comparisonbetween finite and infinite cardinalities. Let a0 ∈ G be any element in G. Studying theedges from a to G \ a, we can write

G \ a = x ∈ G \ a : G |= xRa ∪ x ∈ G \ a : G |= ¬(xRa)

and one of the two sets in the partition must be infinite. Let A0 be such set. We candefine recursively sets A1, . . . , A2k and elements a1, . . . , a2k by choosing ai+1 to be anyelement in Ai and Ai+1 to be an infinite set in the partition of Ai \ a given by twosets, Ai \ ai = x ∈ (Ai \ ai) : G |= xRai ∪ x ∈ (Ai \ ai) : G |= ¬(xRai), fori = 1, . . . , 2k.

By construction, we have A0 ⊇ A1 ⊇ · · · ⊇ A2k, ai+1 ∈ Ai \ Ai+1 for each i < 2k,and furthermore the truth value of the formula (xRai) remains constant for x ∈ Ai+1. Inparticular, for each i, the truth value of R(ai; aj) remains constant for i < j ≤ 2k. Ifthere are indices 0 ≤ j1 < . . . < jk < 2k such that R(ajℓ ; ajℓ+1), for each ℓ, then we obtaina clique given by aj1, . . . , ajk). Otherwise, there are indices 0 ≤ j1 < . . . < jk < 2krealizing ¬R(ajℓ , ajℓ+1) and we obtain an anticlique of size k. This contradicts the factthat G |= ¬P.

1Note that we have not written M |= ¬P , as the “first-order description” of P does not necessarily

imply that P can described by a sentence in the extended language.2This version is equivalent to the version of Ramsey’s theorem with two colors, but the proof given

here can be easily generalized to the case of finitely many colors.

3.2. FURSTENBERG’S CORRESPONDENCE PRINCIPLE 51

3.2 Furstenberg’s correspondence principle

One application of ultraproducts of finite structures is to give a general framework forcorrespondence principles between finite structures and dynamical systems. We knowpresent a proof of the original correspondence argumend given by Furstenberg in [7],following closely the proof given by I. Goldbring and H. Towsner in [10]. Since the firstpaper of Furstenberg, many variations and generalizations have ben produced (see, forinstance, [?],[2]) and the method presented here extends naturally to these generalizations.

Definition 3.2.1. Let E ⊆ Z. The upper Banach density of E is defined as

d(E) = lim supm−n→∞

|[n,m] ∩ E|m− n

= infn∈N

supm≥n

|[n,m] ∩ E|m− n

For example, it is easy to see that d(2N) =1

2and d(P) = 0 if P is the set of prime

numbers. (The second example will be clear after remember that π(x) ∼ x

log xwhen

x→ ∞Definition 3.2.2. A dynamical system is a tuple (Y,B, µ, T ) such that (Y,B, µ) is aprobability space and T : Y −→ Y is a invertible measure-preserving transformation,that is, T : Y −→ Y is a bimeasurable bijection for which µ(A) = µ(TA) for all A ∈ B.

Theorem 3.2.3. (Furstenberg’s correspondence theorem) Let E ⊆ Z with positiveupper Banach density be given. Then there is a dynamical system (Y,B, µ, T ) and a setA ∈ B with µ(A) = d(E) such that for any finite set of integers U ,

d

(⋂

i∈U

(E − i)

)≥ µ

(⋂

i∈U

T−iA

).

Proof. Let 〈ǫN : N ∈ N〉 is a increasing sequence of positive rational numbers such thatsupNǫN = d(E). Then for each N ∈ N there is some n such that

ǫN < supm≥n

|[n,m] ∩ E|m− n

.

That is, there are m,n ∈ N, m− n > N with|[n,m] ∩ E|m− n

≥ ǫN .

Define the functions

fn,m :[n,m] −→ [n,m]

x 7−→ fn,m(x) =

x+ 1 if n ≤ x < m

n if x = m

Now, consider the classical measured structures Mn,m = ([n,m], E∩[n,m], fn,m) equippedwith the normalized counting measure. Let Mn,m be the AML-structure associated toMn,m.

Let (Y,A, T ) =∏

U Mn,m, B = σ(Def1(M) and µ the canonical measure over B.


• (Y,B, µ, T ) is a dynamical system: It is clear by construction that (Y,B, µ) isa probability space. Now, since each fn,m is a bijection in the finite structure([n,m], E ∩ [n,m], fn,m), is an invertible measure-preserving transformation. ByŁoś’Theorem, we obtain that T : Y −→ Y is an invertible measure-preservingtransformation.

By construction, µ(A) = d(E). Let U be a finite subset of Z. We will show that

d

(⋂i∈U

(E − i)

)≥ µ

(⋂i∈U

T−iA

). Let δ be a rational number such that δ < µ(

⋂i∈U T

−i(A)

(if there is no such a δ, there is nothing to show).

By Łoś’ Theorem again, for each N we can find m,n ∈ N, m− n > N such that

|[n,m] ∩ ⋂i∈U

f−in,m(E ∩ [n,m])|

m,n≥ δ.

So, for a fixed γ > 0, and for N sufficiently large, we have that

|[n,m] ∩ ⋂i∈U

(E − i)|

m,n> δ − γ

(Note here that f−in,m(E ∩ [n,m]) ⊆ (E − i) for all i ∈ U .) Therefore,

d

(⋂

i∈U

(E − i)

)= inf

n∈Nsupm≥n

|[n,m] ∩ ⋂

i∈U

(E − i)|

m,n

> δ − γ

Since γ is arbitrary, we get that

d

(⋂

i∈U

(E − i)

)≥ δ

and the conclusion of the theorem holds.

3.3 Szemerédi’s Regularity Lemma

The purpose of this section is to present a proof of the Szemerédi’s Regularity Lemmausing measure-theoretic aspects of the ultraproducts of finite structures.

At first, the regularity lemma was the main technical lemma to prove the more cele-brated Szemerédi’s theorem: every subset of the integers with positive density containsarbitrarily large arithmetic progressions.

Since then, the regularity lemma has played a key role in extremal combinatorics, aswell as in other areas such as number theory, algebra, ergodic theory, etc.

More recently, people have been discovering some improved versions of the regularitylemma in different contexts. For instance, Malliaris and Shelah discovered a version ofthe regularity lemma for stable graphs where there are no irregular pairs, and Tao hasproved the so-called of Algebraic regularity lemma in the context of finite fields with sizesgoing to infinity.

3.3. SZEMERÉDI’S REGULARITY LEMMA 53

Definition 3.3.1. A graph G is a pair G = (V,E) where E is a symmetric subset ofV × V

Definition 3.3.2. Let G = (V,E) be a graph (undirected, simple) and let A,B be twodisjoint subsets of V .

1. The edge density between A,B is the proportion of edges between vertices in A and

vertices in B, that is, d(A,B) =|E ∩ (A×B)

|A| · |B| .

2. For ǫ > 0, we say that the pair (A,B) is ǫ-regular if for all A′ ⊆ A, B′ ⊆ B with|A′| ≥ ǫ|A|, |B′| ≥ ǫ|B| we have |d(A,B)− d(A′, B′)| < ǫ.

Example 3.3.3. Every pair (A,B) is ǫ = 1-regular. If A′ ⊆ A,B′ ⊆ B satisfy |A′| ≥ |A|and |B′| ≥ |B|, then A′ = A,B′ = B and so

|d(A′, B′)− d(A,B)| = |d(A,B)− d(A,B)| = 0 < 1.

Every pair of singletons is ǫ-regular for every ǫ > 0. Let A = a, B = b. If A′ ⊆A,B′ ⊆ B satisfy |A′| ≥ ǫ|A| = ǫ > 0, |B′| ≥ ǫ|B| > 0 then again A = A′ and B = B′.

Example 3.3.4. Put A = a and B = B1 ∪ B2 with |B1| = |B2| and x ∈ B : aEx =B1. Then the pair is ǫ-regular for ǫ > 1

2: notice that

d(A,B) =E ∩ (A× B)

|A| · |B| =|B1|2|B1|

=1

2.

So, if B′ ⊆ B satisfy |B′| ≥ ǫ|B| and |A′| ≥ ǫ|A|, we obtain

|d(A,B)− d(A′, B′)| =∣∣∣∣1

2− d(A,B′)

∣∣∣∣ ≤1

2< ǫ

because d(A′, B′) ∈ [0, 1].

Theorem 3.3.5 (Szemerédi’s Regularity Lemma). For any k ∈ N and ǫ > 0 there isK ≥ k such that the following holds:

Whenever (G,E) is a finite graph, there is n ∈ [k,K] and a partition G = U1 ∪ · ∪Unsuch that ∣∣∣

⋃Ui × Uj : (Ui, Uj) is not ǫ-regular

∣∣∣ ≤ ǫ|G|2

This theorem establishes that every graph can be divided into subsets (and there isan absolute bound on the number of subsets it is divided into) so that the edges betweendifferent subsets behave pseudo-randomly for most of the pairs. (at most an ǫ-fraction ofthe pairs are not ǫ-regular).

Before going into the proof, let us analyze the following important examples:

Example 3.3.6 (Bipartite complete graph). Suppose U,W are disjoint sets of size n, andwe define the graph G with V (G) = U ∪W and E(G) = (u, w) : u ∈ U,w ∈ W.

Then, for every ǫ > 0, the partition given by G = U ∪ V has only regular pairs,with uniform densities d(U, U) = 0, d(U, V ) = 1. If necessary, we can consider furtherpartitions of U and V to decompose G into more sets, still having uniform densities either0 or 1.


b b

b

b

b

b

b

b

......

a1 b1

a2 b2

a3 b3

am bm

U V

⇒

U

U

V

V

Example 3.3.7 (Half-graph). Put G = a1, . . . , am∪b1, . . . , bm with the edge relationgiven by aiEbj if and only if i ≤ j. In this case we can start again with the partitionG = U ∪ V , which will produce pairs (U, U) and (V, V ) with uniform density 0, and ǫ-irregular pairs (U, V ). By partitioning further the sets U and V we get more regular pairsof uniform densities 0 and 1, together with some irregular pairs (The half-blue squares inthe representation below).

This example shows that, in general, irregular pairs will always appear. However,notice that if the partition of G has n equal pieces (partitioning U and V in n/2 equalpieces), then total number of pairs is n2 and the total number of irregular pairs is n. Westill have that the proportion of irregular pairs tends to

limn→∞

# irregular pairs

# pairs= lim

n→∞

n

n2= 0.

b b

b

b

b

b

b

b

......

a1 b1

a2 b2

a3 b3

am bm

U V

⇒

U

U

V

V

3.3. SZEMERÉDI’S REGULARITY LEMMA 55

The following proof of Szemerédi’s Regularity Lemma using ultraproducts is due to I.Goldbring and H. Towsner, and appear in [10].

Proof. Towards a contradiction, assume that the conclusion fails. Then there are k ∈N, ǫ > 0 such that for every K ≥ k, there is a finite graph (GK , EK) with no partitioninto k ≤ n ≤ K satisfying the conclusion of the theorem.

Notice that since any pair of singletons is ǫ-regular, we must have |GK | ≥ K for allK.

We will now choose inductively a suitable language to construct an ultraproduct.First, put L0 = E, a binary relation. Now, suppose that L0 ⊆ · · ·Ln have already beenconstructed. For any two formular φ(x), ψ(x) (at least one not in Ln−1), we add unary

predicates Rϕ,ψ(x) and Sϕ,ψ(x) to the language Ln+1. Finally, put L :=⋃

n<ω

Ln.

Every graph (GK , EK) may be seen as an L-structure by interpreting E as the binaryrelation EK , and supposing ϕ, ψ have been interpreted in GK , we may set U = φ(GK),U ′ = ψ(GK) (note that ϕ and ψ may have parameters).

If the pair (U, U ′) is ǫ-regular, then set Rϕ,ψ = ∅ = Sϕ,ψ. Otherwise, there are V ⊆ Uand V ′ ⊆ V ′ such that |V | ≥ ǫ|U |, |V ′| ≥ ǫ|U ′| and |d(V, V ′)−d(U, U ′)| > ǫ. Put Rϕ,ψ = Van Sϕ,ψ(V

′) in this case.Consider now the ultraproduct of L-structures given by G =

∏U GK , with the canon-

ical product measure on G2 generated by µ(X) = st

( |X||G|

).

Let h := E(χE |B2,1), the conditional expectation of the characteristic function χE withrespect to the σ-algebra B2,1.

Since h can be approximated by simple function, we may write h as

h =∑

i,j

αi,jχCi×Dj + h′,

where ||h′||L2 <ǫ4

4and Ci, Di are L-definable subsets of G.

Let U1, . . . , Un be the atoms of the finite algebra generated by Ci ∪ Dj. Thenwe have a refinement given by h =

∑(i,j)≤n αijχUi×Uj + h′′ with ||h′′||L2 ≤ ||h′||L2, and

U1, . . . , Un is a finite partition of G into definable sets.Let B be the collection of pairs (i, j) such that RUi,Uj 6= ∅ 6= SUi,Uj (i.e., the collection

of irregular pairs), and define

αij =µ(E ∩ (Ui, Uj))

µ(Ui) · µ(Uj)and βij =

µ(E ∩ (RUi,Uj × SUi,Uj)

µ(RUi,Uj) · µ(SUi,Uj).

By Łoś’ theorem, for every (i, j) ∈ B we know that, since RUi,Uj , SUi,Uj where therespective subsets of Ui, Uj contradicting ǫ-regularity, the following inequality holds:

|d(Rij;Sij)− d(Ui, Uj)| ≥ |βij − αij | ≥ ǫ.


Suppose now that µ

⋃

(i,j)∈B

Ui × Uj

≥ ǫ

2, and let B+ = (i, j) : αij−βij ≥ ǫ. Without

loss of generality, we may assume that µ

⋃

(i,j)∈B+

Ui × Uj

≥ ǫ

4, since otherwise we can

carry out a similar argument with B− = B \B+ instead of B+.Again, by Łoś’ theorem, we have µ(Rij) ≥ ǫµ(Ui) and µ(Sij) ≥ ǫµ(Uj). Put Z =⋃

(i,j)∈B+ Rij × Sij . Note that Z ∈ B2,1, so we have

0 =

∫(χE − h) · χZ dµ

=

∫χE − ·χZ − hχZ dµ

=

∫ (χE · χZ −

∑

i,j≤n

αijχUi×UjχZ

)dµ−

∫h′′χZ dµ

which implies

∣∣∣∣∫h′′ · χZ dµ

∣∣∣∣ =∣∣∣∣∣

∫ ∑

i,j≤n

αijχUi×UjχZ − χEχZ dµ

∣∣∣∣∣

=

∣∣∣∣∣∣

∫ ∑

i,j≤n

αijχUij

∑

(k,ℓ)∈B+

χRk,ℓ×Sk,ℓ

−

χE ·

∑

(k,ℓ)∈B+

χRk,ℓ×Sk,ℓ

dµ

∣∣∣∣∣∣

=

∣∣∣∣∣∣

∑

(k,ℓ)∈B+

(∫ [∑

i,j≤n

αij · χUi×Uj · χRk,ℓ×Sk,ℓ − χE · χRk,ℓ×Sk,ℓ

]dµ

)∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣

∑

(k,ℓ)∈B+

∫αk,ℓ · χUk×Uℓ · χRk,ℓ×Sk,ℓ︸︷︷︸

=χRk,ℓ×Sk,ℓ

−χE · χRk,ℓ×Sk,ℓ dµ

∣∣∣∣∣∣∣∣

=

∣∣∣∣∣∣

∑

(k,ℓ)∈B+

(αk,ℓ · µ(Rk,ℓ) · µ(Sk,ℓ)− µ(E ∩ (Rk,ℓ × Sk,ℓ)))

∣∣∣∣∣∣

=

∣∣∣∣∣∣

∑

(k,ℓ)∈B+

(αk,ℓ − βk,ℓ)µ(Rk,ℓ)µ(Sk,ℓ)

∣∣∣∣∣∣

≥ ǫ∑

(k,ℓ)∈B+

µ(RUk,Uℓ) · µ(SUk,Uℓ) ≥ ǫ3∑

(k,ℓ)∈B+

µ(Uk) · µ(Uℓ) ≥ǫ4

4,

but this is a contradiction, because by Cauchy-Schwartz inequality we have∣∣∣∣∫h′′ · χZ dµ

∣∣∣∣ = |〈h′′, χZ〉| ≤ ||h′′||L2 · ||χZ||L2 <ǫ4

4· 1.

3.4. THE ERDÖS-HAJNAL PROPERTY FOR STABLE GRAPHS. 57

It follows then that µ

⋃

(i,j)∈B

Ui × Uj

≤ ǫ

2.

Note that until here we have only found a partition which satisfies the regularityconditions in the infinite model, so we have to pull down this partition into some finitemodel GK , obtaining a contradiction.

By choosing K ≥ n large enough, we may ensure that the following hold:

• Whenever (i, j) ∈ B, RUi,Uj(GK) = ∅.

• µ(Ui(GK)) ≤√2(µ(Ui(G)))

This implies that

µ

⋃

(i,j)∈B

Ui(GK)× Uj(GK)

≤ 2µ

⋃

(i,j)∈B

Ui(G)× Uj(G)

≤ ǫ,

that is, ∣∣∣∣∣∣

⋃

(i,j)∈B

Ui(GK)× Uj(GK)

∣∣∣∣∣∣≥ ǫ|G|2,

contradicting that GK was a counterexample of the existence of an ǫ-regular decomposi-tion.

3.4 The Erdös-Hajnal property for stable graphs.

The Erdös-Hajnal conjecture is an interesting problem in graph theory whose statementcan be easily stated in terms of the normalized pseudofinite dimension described in Section2.2.11. In this lecture we will present the main definitions regarding the Erdös-Hajnal con-jecture, and a proof of the Erdös-Hajnal property due to A. Chernikov and S. Starchenko,presented in [6]. Recall the following definitions:

Definition 3.4.1. Let G be a graph. A clique in G is a set of vertices all pairwiseadjacent, and an anticlique in G is a set of vertices that are all pairwise non-adjacent.

Definition 3.4.2. Let H be a graph. We say that another graph G is H-free if G doesnot contain an induced subgraph isomorphic to H .

One of the first successful applications of the probabilistic method is due to Erdös,who use it to show the following:

Theorem 3.4.3 (Erdös, 1947). For every integer k ≥ 3, the Ramsey number of k satisfiesR(k) > 2k/2. That is, for every k ≥ 3 there is a graph Gk of size with more than 2k/2

vertices such that Gk does not contain neither a clique nor an anticlique of size k.


The corollary of Erdös theorem show that, in general, for a graph of n vertices wecannot expect neither cliques nor anticliques of size around O(logn). However, the waythe probabilistic method was used in its proof suggested that the graphs used as examplessatisfy (in the limit) the same properties as the random graph, so in particular, every graphH can be found with probability 1 in the graphs Gk. Erdös and A. Hajnal conjectured in1989 that the situation was very different for “structured” graphs:

Conjecture 3.4.4 (Erdös-Hajnal conjecture, 1989). For every finite graph H there is aconstant δ = δ(H) > 0 such that every finite H-free graph G = (V,E) contains either aclique or an anticlique of size at least |V (G)|δ.

This conjecture is known to be true for several classes of graphs H , as well as for everygraph of size |V (H)| ≤ 4.

Definition 3.4.5. For m ∈ N let Hm be the half-graph on 2m vertices, i.e., Hm is abipartite graph whose vertex set is a disjoint union of a1, . . . , am ∪ b1, . . . , bm suchthat aiEbj if and only if i ≤ j.

b b

b

b

b

b

b

b

......

a1 b1

a2 b2

a3 b3

am bm

The following theorem is a corollary of ([23], Theorem 3.5)

Definition 3.4.6. For every m ∈ N, we say that a graph G is Hm-free∗ if G does notcontain Hm as a subgraph.

Remark 3.4.7. It is important to notice the subtle difference with the concept of anHm-free graph. The main difference, is that an Hm-free∗ graph does not have to avoidHm only as an induced subgraph, but rather as a subgraph. That is, it cannot containvertices a1, . . . , am, b1, . . . , bm with the edge relation aiEbj ⇔ i ≤ j, regardless the edgerelation inside the sets a1, . . . , am and b1, . . . , bm. In other words, to be Hm-free∗,it is necessary to avoid a finite family of graphs (namely, all graphs with the m-orderproperty).

The following theorem is a striking example that model-theory can be meaningful inthis contexts and shed some lights on the Erdös-Hajnal property for different classes ofgraphs.


Theorem 3.4.8 (Chernikov-Starchenko, 2015). For every m ∈ N there is a constantδ(m) > 0 such that every finite graph that does not contain Hm as a subgraph G = (V,E)contains either a clique or an anticlique of size at least |V |δ(m).

Remark 3.4.9. This is not the same as proving the Erdös-Hajnal conjecture for thegraph Hm. Rather, it shows that if a graph G does not have the m-order property for theformula xRy, then it contains either a clique or an anticlique of size at least |V |δ(m).

In what follows, we will present a proof of Theorem 3.4.8 following closely the presen-tation given in [6].

Proposition 3.4.10. Let Y ⊆M×Mm and Z ⊆M definable. Assume that δ(Z) = α andfor all pairwise distinct a1, . . . , am ∈ Z, we have δ(x ∈ M : (x, a1, . . . , am) ∈ Y ) ≤ β.Then,

δ

(x ∈M : ∃z1, . . . , zm ∈ Z

(∧

i 6=j

zi 6= zj ∧ (x, z1, . . . , zm) ∈ Y

))≤ m · α + β

Proof. Consider Z(m) = (z1, . . . , zm) ∈ Zm : z1, . . . , zm are pairwise different and define

Y ′ := (x, a) ∈ Y : a ∈ Z(m).

First, let f : Y ′ →M be the projection on the first coordinate. Then we have,

f(Y ′) =

x ∈ M : ∃z1, . . . , zm ∈ Z

(∧

i 6=j

zi 6= zj ∧ (x, z1, . . . , zm) ∈ Y

).

On the other hand, we can consider the function g : Y ′ → Z(m) to be the projectionon the last m coordinates. Notice that for every a ∈ Z(m), there is a bijection betweeng−1(a) and x ∈M : (x, a) ∈ Y , and by hypothesis we have

δ(g−1(a) = δ(x ∈M : (x, a) ∈ Y ) ≤ β

for all a ∈ Z(m).So, by Proposition 2.2.7, we have that

δ(f(Y ′)) ≤ δ(Y ′) ≤ δ(Z(m)) + β

≤ δ(Zm) + β = m · δ(Z) + β

= m · α + β.

Theorem 3.4.8 is implied by the following non-standard version:

Theorem 3.4.11. Let V be a pseudofinite set and E ⊆ V × V a definable symmetricsubset. Assume that the graph (V,E) is Hm-free* for some m ∈ N. Then there is adefinable A ⊆ V such that δ(A) > 0 and either (a, a′) ∈ E for all a 6= a′ in A, or(a, a′) 6∈ E for all a 6= a′ in A.


We know show that this result implies Theorem 3.4.8:

Proof. Assume 3.4.8 is false. Then, there is m ∈ N such that for every n < ω there is aHm-*-free graph Gn such that Gn does not contain neither a clique nor an anticlique ofsize |V (Gn)|

1

n .Let U be a non-principal ultrafilter on ω, and put G =

∏U Gn. We have:

1. G is infinite: Notice that any pair of points in Gn is either a clique or an anticlique.So we have 2 < |Gn|1/n which implies 2n < |Gn| for any n < ω, and limn→U |Gn| =+∞.

Note also that G is Hm-*-free graph, because of Łoś’ theorem and the fact that

Gn |= ¬∃x1, . . . , xm, y1, . . . , ym(∧

i<j≤m

xi 6= xj ∧ yi 6= yj ∧∧

i≤j

xEyj ∧∧

i>j

¬xiEyj).

By Theorem 3.4.11, there is a definable subset A ⊆ V with δ(A) > 0, which is eithera clique or an anticlique. Let us write A =

∏U An, and say δ(A) = ǫ > 0.

Assume without loss of generality that A is a clique. Then, we have

G |= ∀x, y (x 6= y ∧ A(x) ∧A(y) → xEy) .

By Łoś’ theorem, this means that for U-almost all n < ω,

Gn |= ∀x, y (x 6= y ∧An(x) ∧ An(y) → xEy) ,

which means that An ⊆ Gn is a clique for U-almost all n.By the choice of the graphs Gn, we have that |An| ≤ |Gn|

1

n , which implies thatlog |An|log |Gn|

≤ 1

n. So,

δ(A) = limn→U

log |An|log |Gn|

≤ limn→U

1

n= 0 < δ(A),

a contradiction.

We will see a more general proof of Theorem 3.4.11 that works for hypergraphs. As-sume throughout the rest of this section that V =

∏U Vi and E =

∏U Ei is a definable

subset of V n, which is symmetric (i.e., closed under permutation of the coordinates.).

Definition 3.4.12.

1. For v1, . . . , vn−1 ∈ V and a subset X ⊆ V , we define

E(v1, . . . , vn−1, X) := x ∈ X : V |= E(v1, . . . , vn, x).

2. A partitioned formula is a formula φ(x1, . . . , xk; y1, . . . , yℓ) with two distinguishedgroups of variables x, y. We often refer to the variables y as the parameter variables.


3. A partitioned formula φ(x, y) is stable if the graph (V |x|×V ℓ, R) with R := (a, b) ∈V |x| × V |y| : V |= φ(a, b) is Hm-*-free for some m ∈ N.

That is, for some m there are no tuples a1, . . . , am ∈ V k, b1, . . . , bm ∈ V ℓ such thatV |= φ(ai, bj) if and only if i ≤ j.

4. We say that a definable set X ⊆ V is large if δ(X) > 0, and we say that X is smallif δ(X) ≤ 0.

We will use some basic local stability such as definability of types, and Shelah’s 2-rank.We will discuss these as we go through the proof of the following result:

Proposition 3.4.13. Assume that E(x1; x2, . . . , xn) is stable. Then there is a definableset A ⊆ V such that δ(A) > 0 and either (a1, . . . , an) ∈ E for all pairwise distincta1, . . . , an ∈ A, or (a1, . . . , an) 6∈ E for all pairwise distinct a1, . . . , an ∈ A.

Given the assumption of stability and symmetry of E and basic properties of stableformulas, such as:

• Closure of stability under boolean combinations.

• For every stable formula φ(x, y), all φ-types over models are defined by booleancombinations of instances of ψ(y : x) := φ(x; y).

Claim: We can choose a finite set of partitioned formulas ∆ such that:

1. E(x1, . . . , xn) ∈ ∆.

2. ∆ is closed under negation and permutation of variables.

3. If φ(x1, . . . , xs) ∈ ∆ then φ(x1; x2, . . . , xs) is stable.

4. If φ(x; y) ∈ ∆, then every φ-type in x over V is defined by an instance of someformula in ∆.

Proof. Take ∆ = E(x1, . . . , xn),¬E(x1, . . . , xn), then: (1) is clear, (2) follows from thesymmetry of E, and (3) follows from the assumption that E(x1; x2, . . . , xn) is stable.

Finally, since E(x1; x2, . . . , xn) is stable, every E-type has a definition of the form

∨

I⊆1,...,2k,|I|=k

(∧

i∈I

E(ai; x2, . . . , xm)

)

for some a1, . . . , a2k ∈ V .

As every formula in ∆ is stable, R∆(x = x) is finite. Let S ⊆ V be a large definablesubset of the smalles R∆-rank (among large subsets): since R∆(x = x) is finite, we cansimply take

ℓ = minR∆(S) : S ⊆ V, Slargeand pick S ⊆ V such that R∆(S) = ℓ.


By Proposition 2.2.7(4), S cannot be covered by finitely many definable sets of smallerR∆-rank: If S1, . . . , Sk ⊆ S have R∆-rank smaller than ℓ, then S1, . . . , Sk are small (bythe minimality of ℓ) and we would have

δ

(k⋃

i=1

Sk

)= maxδ(Si) : i ≤ k = 0 < δ(S)

So,

k⋃

i=1

Si does not cover S.

Hence, by compactness, there is a complete ∆-type p over V such that R∆(S∪p) =R∆(S): Let π(x) := S ∪ ¬ψ : R∆(ψ ∧ S) < ℓ. By the previous statement, π(x) isconsistent, and any completion p of π(x) will satisfy Rδ(S ∪ p) = R∆(S).

As p is definable, say using parameters from some countable V0 ⊆ V , for every k wehave a complete well-defined type p(k)(x1, . . . , xk) over V given by

p(k)(x1, . . . , xk) =⋃

tp∆(ak, . . . , a1/V′) : V0 ⊆ V ′ ⊆ V , V ′ is countable, and ai+1 |= p V ′a0...ai for i

We know show the type p(k) is well-defined: Let a1, . . . , ak and a′1, . . . , a′k be tuples such

that ai+1 |= p V ′a1...ai and a′i+1 V ′a′1...a′i

for i < k. If φ(x1, . . . , xk; b) ∈ tp∆(a1, . . . , ak), wehave:

φ(x1, . . . , xk; b) ∈ tp∆(a1, . . . , ak)

⇒ φ(a1, . . . , ak−1, xk; b) ∈ p V ′a1,...ak−1

⇒ (a1, . . . , ak−1, b) |= dpφ(xk; x1, . . . , xk−1, y) ∈ pk−1

⇒ (a1, . . . , ak−1) |= dpφ(xk; x1, . . . , xk−1, b)

⇒ (a′1, . . . , a′k−1) |= dpφ(xk; x1, . . . , xk−1, b) (Induction hypothesis)

⇒ dpφ(xk; a′1, . . . , a

′k−1, b) ∈ p V ′a′

1...a′

k−1

⇒ φ(x1, . . . , xk; b) ∈ tp∆(a′1, . . . , a

′k).

It is clear that p(k) is a complete k-type.Claim: For any formula r(x1, . . . , xk) ∈ ∆, if p(k) ⊢ r(x1, . . . , xk) then there is a large

definable subset A ⊆ S such that |= r(a1, . . . , ak) holds for any pairwise distinct a1, . . . , akfrom A.

Proof. We prove this by induction on k. For k = 1, suppose p(x) ⊢ r(x1) for r(x1) ∈ ∆,and take A = r(S). It is necessary only to show that A is large.

By the choice of p, the R∆-ranks of S and S ∧ r are equal. Therefore, R∆(S ∧ ¬r) <Rδ(S), and by the minimality of R∆(S) among large sets, we have that δ(S ∧ ¬r) = 0.

Thus, δ(A) = δ(S ∧ r) = δ(S) > 0, and we conclude that A is large.Assume now the result for k > 1. By the choice of ∆, there is some formula

ψ(x1, . . . , xk−1) ∈ ∆ such that the type p r(x1,...,xk−1;xk) is defined by ψ(x1, . . . , xk−1),that is,

r(v1, . . . , vk−1; xk) ∈ p(xk) ⇔ V |= ψ(v1, . . . , vk−1).


By induction hypothesis, there is a large definable set B ⊆ S such that V |= ψ(b1, . . . , bk−1)for all pairwise distinct b1, . . . , bk−1 ∈ B.

As B is definable, B can be written as B =∏

U Bi for some Bi ⊆ Si. For each i < ω,let Ai ⊆ Bi be maximal under inclusion such that r(a1, . . . , ak) holds for all pairwisedistinct a1, . . . , ak from Ai, and let A =

∏U Ai. We have:

(i) A ⊆ B.

(ii) V |= r(a1, . . . , ak) for any pairwise distinct a1, . . . , ak ∈ A.

(iii) For any b ∈ B \ A there are some pairwise distinct a1, . . . , ak−1 ∈ A such thatV |= ¬r(a1, . . . , ak−1, b). This follows from the maximality of Ai ⊆ Bi, and by Łoś’Theorem.

We will show that A is large. In fact, we will show that δ(A) ≥ 1

k − 1δ(B).

Assume otherwise, and say δ(A) = α1 <1

k − 1δ(B). Since A ⊆ B, we have that for

all pairwise distinct a1, . . . , ak−1 ∈ A, V |= ψ(a1, . . . , ak−1). So, r(a1, . . . , ak−1; xk) ∈ p.By the choice of p, the R∆-rank of S(xk) ∧ r(a1, . . . , ak−1; xk) is equal to the R∆-rank

of S, so the R∆-rank of S(xk) ∧ ¬r(a1, . . . , ak−1; xk) is smaller than R∆(S).This implies that B \ r(a1, . . . , ak−1;S) is small by the choice of S, that is, δ(B \

r(a1, . . . , ak−1;B)=0. By condition (iii), we have

B \ A ⊆⋃

a1,...,ak−1∈Adistinct

(B \ r(a1, . . . , ak−1;B)

and by Proposition 2.2.7(6), δ(B \ A) ≤ (k − 1)δ(A) + 0 = (k − 1)α1. This implies byProposition 2.2.7(4)

δ(B) = maxδ(A), δ(B \ A) ≤ maxα1, (k − 1)α1 ≤ (k − 1)α1 < δ(B),

a contradiction.

Finally, as both E(x1, . . . , xn) and ¬E(x1, . . . , xn) are in ∆ and p(n) is complete, eitherp(n) ⊢ E(x1, . . . , xn) or p(n) ⊢ ¬E(x1, . . . , xn). This concludes the proof of Theorem 3.4.13.The case n = 2 is the Theorem 3.4.11


Lecture 4

Strongly minimal pseudofinite

structures

Strongly minimal sets control uncountably categorical theories and more generally ω-stable theories of finite Morley rank. The model theory of strongly minimal sets is verywell-known and is the most accesible special case of general stability theory.

The basic examples of strongly minimal structures are algebraically closed fields (inthe ring language), infinite vector spaces over division rings, and infinite free G-sets inthe language L = g · ∗ : g ∈ G.

In this chapter we will present the following result, which is presumably a folkloreresult that can be traced back to Zilber. The proof we present here follows the expositionof [24].

In the first section we will collect some standard definitions and results about stronglyminimal structures. The reader with background in this subject can simply skip to Section4.2 where these results are applying for pseudofinite structures.

4.1 Strongly minimal structures

Definition 4.1.1. A compete 1-sorted theory T in a language L is said to be stronglyminimal if every definable subset X ⊆ M1 (possibly defined with parameters) of anymodel M |= T is either finite or cofinite.

For this chapter, we fix a complete strongly minimal L-theory T , a saturated modelD of T . The cartesian powers of D are the sets Dn for n ≥ 1, and we fix an auxiliarypoint D0 = ∗.Definition 4.1.2.

1. Let X ⊆ Dn be a definable set. Then dim(X) is the least k ≤ n such that we canwrite X as a finite union of definable sets X1 ∪ · · · ∪Xr such that for each i thereis a projection πi : D

n → Dk such that πi Xi: Xi → Dk is finite-to-one.

2. Let X ⊆ Dn be a definable set of dimension k. Then mult(X) is the greatest naturalnumber m (if one exists) such that X can be written as a disjoint union of definablesets X1, . . . , Xm such that dim(Xi) = k for each i.

65

66 LECTURE 4. STRONGLY MINIMAL PSEUDOFINITE STRUCTURES

3. A k-cell is a definable set X ⊆ Dn for some n ≥ k such that for some r 6= 0 thereis a projection π : Dn → Dk such that dim(π(X)) = k and π X is r-to-1.

Remark 4.1.3. Clearly dim(X) as defined in (1) exists, because the projection identityπ : Dn → Dn is one-to-one.

Lemma 4.1.4. If X = X1 ∪ · · · ∪Xm are all definable sets, then dim(X) = dim(Xi) forsome i ≤ m.

Proof. SupposeX ⊆ Dn. Note that if for some Y ⊆ Dn we have a projection π : Dn → Dk

such that π|Y is finite-to-one, then whenever k ≤ k′ ≤ n there is a projection π : Dn → Dk′

that is also finite-to-one, simply by adding more variables on which take the projection(that is, if π−1|Y (a) is finite, certainly π−1|Y (a, b1, . . . , bk′−k) is finite too).

Suppose now that dim(Xi) < dim(X) for all i ≤ m. Consider for every i ≤ m thedecomposition Xi = Y 1

i ∪ · · · ∪ Y rii with projections πji : Y

ji → Ddim(Xi) which are finite-

to-one, witnessing that Xi has dimension dim(Xi). We can adjust these projection tofind other projections πji : Dn → Ddim(X)−1 such that πij |Y ji is finite-to-one, obtaining

a decomposition X =⋃

1≤i≤m,1≤j≤ri

Y ji together with projections πji showing that X has

dimension less than or equal to dim(X)− 1. A contradiction.

Proposition 4.1.5.

1. mult(X) exists for any definable X.

2. For any n ≥ 0, Dn has dimension n and multiplicity 1.

3. Any k-cell has dimension k. Moreover, any definable set X is a finite disjoint unionof cells, i.e., k-cells for possibly varying k.

4. For X ⊆ Dn definable, dim(X) = 0 if and only if X is finite.

Proof.

4. Recall that D0 is an auxiliary point D0 = ∗. Then, X is finite if and only if themap π : X → ∗ has finite domain iff π = π0|X is finite-to-one, (when π0 is theconstant projection Dn → D0) iff dim(X) = 0.

2. By induction on n. For n = 0, D0 = ∗ is finite and by (4) we have that dim(D0) =0. Also, since D0 has a single point, if D0 is written as a disjoint union of non-emptysets, the union has to contain only one set. Thus, mult(D0) = 1.

For n = 1, notice that X = D1 ⊆ D1, so dim(D1) ≤ 1 as witnessed by the identityprojection π : D1 → D1. Now, since D1 is saturated, in particular it is infinite, andso dim(D1) 6= 0 by (4). So, dim(D1) = 1.

For the multiplicity, let us assume that mult(D1) = m ≥ 2. Then there are disjointdefinable sets X1, . . . , Xm such that D1 = X1 ∪ · · · ∪ Xm and dim(Xi) = 1 fori = 1, . . . , m. Consider the definable set Y = X2 ∪ · · · ∪ Xm. Since D is strongly

4.1. STRONGLY MINIMAL STRUCTURES 67

minimal, either Y ⊆ X2 or Y c = X1 is finite, but this is a contradiction becauseboth X1, X2 have dimension 1, and in particular they are infinite by (4).

Thus, we conclude that mult(D1) = 1.

Now, we assume as induction hypothesis that dim(Dn) = n and mult(Dn) = 1.Suppose for a contradiction that dim(Dn+1) ≤ n. Then there are definable setsX1, . . . , Xm ⊆ Dn+1 and projections πi : D

n+1 → Dk so that k ≤ n, πi|Xi is finite-to-one and Dn+1 = X1 ∪ · · ·Xm.

Given a ∈ Dn, we can define the set ℓa := (a, y) : y ∈ D (“the line above a).Given Xi, we can consider the definable set Γi = y ∈ D : (a, y) ⊆ D1. By strongminimality, Γi is either finite or cofinite. If finite, then Xi ∩ ℓa is finite. If Γi iscofinite, then ℓa \Xi is finite.

So, we have showed that for every Xi either Xi∩ℓa is finite or ℓa\Xi is finite, and weconclude that every ℓa is “almost contained” in at least one set Xia with 1 ≤ ia ≤ m.

(i.e., ℓa \Xia is finite for some index 1 ≤ ia ≤ m).

Notice that if ℓa is “almost contained” in Xi, then the projection πi sends the vari-ables (x1, . . . , xn, xn+1) to a tuple of variables (xi1 , . . . , xik−1

, xn+1) with 1 ≤ i1 <· · · < ik−1 ≤ n, since otherwise the map πi|Xi would not be finite-to-one.

Consider now the sets Yi := a ∈ Dn : ℓa is almost contained in Xi. The sets Yiare definable, because of the uniform bound of |ℓa \Xi| while a varies. We then havethat Dn = Y1 ∪ · · · ∪ Ym, and we can consider the projections πi = π πi, where πis the projection forgetting only the last variable.

The maps πi are projections from Dn to Dk−1, and πi|Yi = (π πi)|Xi are finite-to-one. Since dim(Dn) = n, one of the projections πi uses n variables, and so thecorresponding projection πi : D

n+1 → Dk uses n + 1 variables. So, k = n + 1. Acontradiction.

e now show that mult(Dn+1) = 1. Assume Dn+1 = X1 ∪ X2, where X1, X2 aredisjoint non-empty definable subsets of Dn+1 of dimension n+ 1. Define Yj = a ∈Dn : ℓa is almost contained in Xj for j = 1, 2. Then Dn = Y1 ∪ Y2 and Y1, Y2 aredisjoint, which implies that one of them must have dimension lower than n. Assumewithout loss of generality that dim(Y2) ≤ n− 1.

If Y2 = Z1 ∪ · · · ∪ Zm is a decomposition witnessing that dim(Y2) = k < n, withprojections πi : D

n → Dk, then

X2 = (Z1 ×D) ∪ · · · ∪ (Zm ×D) ∪ π−1|X2(Y1),

with π being the projection on the first n coordinates, is a decomposition showingthat dim(X2) ≤ n, a contradiction.

Thus, mult(Dn+1) = 1, and we conclude the proof of (2).

1. Let X ⊆ Dn be a definable set with dim(X) = k, and suppose there are infinitelymany disjoint sets (Xi : i < ω) such that dim(Xi) = k and Xi ⊆ X for every i < ω.


Since dim(X) = k, there are definable sets Y1, . . . , Yr and projections πj : Dn → Dk

such that X = Y1 ∪ · · · ∪ Yr and πj Yj is finite-to-one. Then, for every i < ω, we

can consider the sets Xji := Xi ∩ Yj for j = 1, . . . , r. By Lemma 4.1.4, for every Xi

there is a unique minimal index ji ≤ r such that Xi ∩ Yji has the same dimensionas Xi (that is, dim(Xi ∩ Yji) = k).

By the piggeonhole principle, there are infinitely many indices i < ω such thatXi ∩ Yj has dimension k, for a fixed j ≤ r, and by restricting our attention to thoseindices we have that the following:

• The projection πj : Dn → Dk satisfies that πj |Yj is finite-to-one.

• dim(Xi ∩ Yj) = k for all i < ω.

• Xi ∩Xi′ = ∅ for all i 6= i′.

It is clear that for every i < ω, πj(Xi∩Yj) has dimension k, since otherwise we wouldbe able to extend the projections πj to projections from Dn witnessing dim(Xi ∩Yj) < k. For simplicity, let us write π = πj , Y = Yj and Xi = Xi ∩ Yj .Since mult(Dk) = 1 (by (4)), there are not disjoint subsets Z1, Z2 of Dk such thatdim(Z1) = dim(Z2) = k. Moreover, if Z1, Z2 are subsets of Dk of dimension k, thendim(Z1 ∩ Z2) = k.

Consider the type p(y) := ∃x(Xm(x) ∧ π(x) = y) : m < ω. This type is finitelyconsistent because dim

(⋂i≤m π(Xi)

)= k. So, by saturation of D, there is a ∈ D

realizing p(y), and this implies that π−1(a)∩Xi 6= ∅ for all i < ω, and since the setsXi are disjoint, we conclude that π−1(a) is infinite. This contradicts the fact that πis finite-to-one.

3. Let X be a k-cell, that is, X ⊆ Dn and for some r 6= 0 there is a projectionπ : Dn → Dk such that dim(π(X)) = k and π|X is r-to-1.Then, π and Y1 = X serveas a decomposition that shows that dim(X) = k.

Now let X be an arbitrary definable set of dimension k. Note that if X = Y1∪· · ·∪Yris a union of disjoint sets of dimension k, then by decomposing each Yi into cells weobtain a decomposition of X. So, we may assume without loss of generality thatmult(X) = 1.

Let X = Y1 ∪ · · ·Yℓ and πi : Yi → Dk be finite-to-one projections. By usingintersections and complements, and possibly repeating projections, we may assumethat Y1, . . . , Yℓ are disjoint. By reordering, we can also assume that dim(X) =dim(Y1) and dim(Yi) < dim(X) for i = 2, . . . , m. By induction on dim(X), we mayalso assume that every Yi can be written as a union of cells.

Consider for r < ω the set Y1,r = y ∈ Y : |π−1i (πi(y))| = r. By strong minimality,

there is r < ω such that Y1,t = ∅ for t > r. Then we have the disjoint unionY1 = Y1,1 ∪ · · · ∪ Y1,r, and since dim(Y1) = k, one of the sets mut have dimension k(say Y1,s) and the rest have dimension less than k.

So, since πi : Y1,1 → πi(Y1,1) is s-to-one, we have that πi(Y1,s) ⊆ Dk with dim(πi(Y1,s)) =RM(πi(Y1,s)) = RM(Y1,s) = k. Thus, Y1,s is itself a k-cell, and the result follows.


The main point is to deduce “good behavior” for definable sets in higher ambientspaces, using some basic lemmas about algebraic closure in strongly minimal sets.

For example, for b1, . . . , bn ∈ D and A a small subset of D, we say that b1, . . . , bn isalgebraically independent over A if bi ∈ acl(A, b1, . . . , bi−1, bi+1, . . . , bn) for each i. This isequivalent to bi ∈ acl(A, b1, . . . , bi−1) for all i = 1, . . . , n, by the exchange property.

Proposition 4.1.6 (Exchange principle). Suppose D is strongly minimal, A ⊆ D andb, c ∈ D. If b ∈ acl(Ac) \ acl(A), then c ∈ acl(Ab).

Proof. Let φ(x, a, c) be the algebraic formula witnessing that b ∈ acl(Ac), and assume that|φ(D, a, c)| = n. Consider the formula ψ(y) := ∃=nx(φ(x, a, y). We have that D |= ψ(c).If φ(b, a, y) ∧ ψ(y) defines a finite set, this would imply that c ∈ acl(Ab) and we aredone. Otherwise, by strong minimality, |D \ (φ(b, a,D) ∧ ψ(D))| = ℓ for some ℓ ∈ N.Consider now the formula χ(x) := ∃=ℓy(¬(φ(x, a, y) ∧ ψ(y)). Notice that D |= χ(b), sosince b 6∈ acl(A) we have that χ(x) defines an infinite set in D.

Let b1, . . . , bn+1 be different elements in χ(D). We have that φ(bi, a,D) ∧ ψ(D) iscofinite for each 1 ≤ i ≤ n+1, and so we can take c′ ∈ D such that c′ |= ∧n+1

i=1 (φ(bi, a, y)∧ψ(y)), which is a contradiction because c′ |= ψ(y) implies n = |φ(D, a, c′)| ≥ |b1, . . . , bn+1| =n+ 1.

Therefore, we conclude that c ∈ acl(Ab).

Proposition 4.1.7. A set b1, . . . , bn is algebraically independent over A if and only ifbi 6∈ acl(A, b1, . . . , bi−1) for all i = 1, . . . , n.

Proof. The direction (⇒) is clear by definition. Suppose now that b1, . . . , bn are notalgebraically independent over A, and let i ≤ n minimal such that

bi ∈ acl(A, b1, . . . , bi−1, bi+1, . . . , bn).

If i = n, or bi ∈ acl(A, b1, . . . , bi−1), we are done. Otherwise, let j ≥ i minimal suchthat bi ∈ acl(A, b1, . . . , bi−1, bi+1, . . . , bj+1), and put A = A ∪ b1, . . . , bi−1, bi+1, . . . , bj.By minimality, bi ∈ acl(Abj+1) \ acl(A), so by the exchange principle, bj+1 ∈ acl(Abi) =acl(A, b1, . . . , bi−1, bi, bi+1, . . . , bj).

Definition 4.1.8. For b a tuple from D, dim(b/A) denotes the cardinality of some (equiv-alently any) maximal subtuple of b algebraically independent over A.

Proposition 4.1.9. For a subset X ⊆ Dn, definable over A,

dim(X) = maxdim(b/A) : b ∈ X.

Proof. By induction on n. If X = φ(D, a) is an A-definable subset of D1, we have bystrong minimality that X is either finite or cofinite. If X is finite, then for every b ∈ Xwe have b ∈ acl(a) ⊆ acl(A), so maxdim(b/A) : b ∈ X = 0 = dim(X).

If X is cofinite, consider the type

p(x) = φ(x, a) ∪ ¬ψ(x) : ψ(x) ∈ L(A) and ψ(D) is finite.


This type is finitely consistent in D, since the finite union of finite sets is finite, and bysaturation there is b |= p(x). Note that b ∈ X \ acl(A), so

1 = maxdim(b/A) : b ∈ X

On the other hand, since X ∪ (Xc) = D1 and Xc is finite, we have that dim(D1) = 1 =maxdim(X), dim(Xc) = maxdim(X), 0 = dim(X).

Now we proceed with the induction step. Let X = φ(Dn+1, a) be an A-definable subsetof Dn+1. If dim(X) = n+ 1, consider the type

p(x) := φ(x, a) ∪ ¬ψ(x) : ψ(x) ∈ L(A) and dim(ψ(Dn+1) < n+ 1.

If Γ0 ⊆ φ(x, a ∪ ¬ψi(x) : i ≤ m is a finite subset of p(x), then φ(x, a) ∧∧i≤m ¬(ψ(x)) is non-empty, since otherwise we would have φ(x, a) ⊢ ∨i≤m ψi(x) which

implies dim(X) = dim(φ(Dn+1, a)) ≤ maxdim(ψi(Dn+1)) : i ≤ m ≤ n, a contradiction.

By saturation, there is a tuple b = (b1, . . . , bn+1) ∈ Dn+1 realizing the type p(x). Itremains to show that b1, . . . , bn+1 is an A-independent set. Suppose not, and let kbe minimal such that bk+1 ∈ acl(A, b1, . . . , bk), begin ψ(y, a, b1, . . . , bk) be an algebraicformula realized by bk+1.

Consider the formula

ψ(x1, . . . , xn+1; a) := ψ(xk+1; a, x1, . . . , xk).

We have that ψ(D) ⊆ Dn+1, and we can consider the projection π : Dn+1 → Dn thatomits the (k + 1)-th component. Since ψ(xk+1; a, x1, . . . , xk) is an algebraic formula, themap π|ψ(D) is finite-to-one, which implies that dim(ψ) ≤ n. However, this contradicts the

fact that b |= p(x). This contradiction allows us to conclude that maxdim(b/A) : b ∈X = n + 1 = dim(X).

Suppose now that dim(X) ≤ n. Then there is a decomposition X = Y1 ∪ · · · ∪ Yℓinto disjoint cells, with dim(Y1) = dim(X). Since Y1 is a cell, there is a projectionπ : Dn+1 → Ddim(X) such that dim(π(X)) = dim(X) and π|Y1 is r-to-1. By induction

hypothesis, there is b′ ∈ π(X) such that dim(X) = dim(π(X)) = dim(b

′/A).

Let b be an element in Y1 such that π(b) = b′. Since b

′ ∈ Ddim(X) and dim(b′/A) =

dim(X), we also know that the tuple b′is algebraically independent over A. So, since b

′

is a subtuple of b, we have dim(b/A) ≥ dim(b′/A) = dim(X).

On the other hand, since b satisfies the algebraic formula “y ∈ Y1 ∧ π(y) = b′”, we also

have dim(b/A) ≤ dim(b′/A) = dim(X).

What we have called dimension is the same thing as Morley rank (or RM), and whatwe called multiplicity is the same thing as Morley degree (or dM), all in the special caseof strongly minimal theories.

Below, we recall the main definition and results about Morley rank and Morley degree.

Definition 4.1.10 (Morley rank, Morley degree). 1. For a definable set X ⊆ Dn, wedefine inductively on the ordinal α the relation RM(X) ≥ α as follows:


• RM(X) ≥ 0 if and only if X 6= ∅.• If α is a limit ordinal, RM(X) ≥ α if and only if RM(X) ≥ β for all β < α.

• RM(X) ≥ α+ 1 if and only if there is a family of disjoint definable subsets ofX (Xi : i < ω) such that RM(Xi) ≥ α for every i < ω.

2. We say that the Morley rank of X is α if RM(X) ≥ α but RM(X) 6≥ α + 1.

3. If RM(X) = α, we define the Morley degree of X as

dM(X) := maxn < ω : There are X1, . . . , Xn ⊆ X, pairwise disjoint, of Morley rank α

Definition 4.1.11 (Morley rank for types). 1. Let p ∈ Sn(A). Then RM(p) := infRM(X) :X ∈ p and dM(p) := dM(X) : X ∈ p and RM(X) = RM(p).

2. If a ∈ Dn, we define RM(a/A) := RM(tp(a/A)).Note that from these definitions it is clear that given a type p there is a formula φp ∈ p

such that (RM(p), dM(p)) = (RM(φp), dM(φp)).

Lemma 4.1.12. If X ⊆ Dn is A-definable, then RM(X) = supRM(a/A) : a ∈ X.

Proof. By definition, RM(a/A) ≤ RM(X) for all a ∈ X. Thus, supRM(a/A) : a ∈X ≤ RM(X). conversely, we can consider the type p(x) = X(x)∪¬φ(x) : RM(φ) <RM(X).

If Γ ⊆ X,¬φ1, . . . , φn is a finite subset of Γ, then we have that

RM

(n∨

i=1

φi

)= maxRM(φi) : 1 ≤ i ≤ n < RM(X),

and therefore X cannot be contained in⋃ni=1 φi(D). That is, there is bΓ |= Γ.

By saturation, there is a ∈ Dn such that a ∈ p(x), and by construction, we will havethat RM(a/A) = RM(X)

Lemma 4.1.13. If b ∈ acl(A, a), then RM(a, b/A) = RM(a/A).

Proof. Clearly RM(a, b/A) ≥ RM(a/A). We know prove by induction on ordinals thatRM(a, b/A) ≥ α implies RM(a/A) ≥ α.

For α = 0 or for α a limit ordinal, this is clear. Suppose now that RM(a, b/A) ≥α + 1. Let θ(x) ∈ tp(a/A) be a formula such that RM(a/A) = RM(θ(x)) = α anddM(φ) = 1, and let ψ(y; a) ∈ tp(b/Aa) be a formula such that |ψ(y; a) = r, with rminimal. We consider now the formula φ(x, y) := θ(x) ∧ ψ(y; x) ∧ ∃=ry(ψ(y, x)). Sinceφ(x, y) ∈ tp(a, b/A), we know that RM(φ(x, y)) ≥ RM(a, b/A) ≥ α + 1.

Let (Xi : i < ω) be a collection of pairwise disjoint definable subsets of φ(Dn+1)contained in φ(x, y), with RM(Xi) ≥ α, and consider the formula χi(x) := ∃y(Xi(x, y)).

1. RM(χi(x)) ≥ α: since Xi has Morley rank at least α, there are c, d such that|= Xi(c, d) and RM(c, d) ≥ α. By induction hypothesis, since d will b algebraicover Ac, we have that RM(c) ≥ α, and so, RM(χi(x)) ≥ RM(c/A) ≥ α.


2. RM(χ1 ∧ · · · ∧ χm) ≥ α for all m: If not, let m be minimal such that RM(χ1 ∧· · · ∧ χm) < α. Then, RM(χ1 ∧ · · · ∧ χm−1) = α = RM(χm), and since we have

χm =

(χm ∩

m−1⋂

i=1

χi

)∪(χm ∩ ¬

m−1⋂

i=1

χi

)

and the first part of the disjoint have rank less than α, we have thatRM(χm ∧ ¬∧m−1

i=1 χi)≥

α.

But then, both∧m−1i=1 χi and ¬∧m−1

i=1 χi have rank at least α, and are included inθ(x), contradicting the fact that dM(θ).

Since D is saturated, there exists c ∈ D such that χi(c) for all i < ω. So, for each i, thereis di such that χi(c, di) holds, but all those elements have to be different because the setsXi are disjoint. Also, χi(c, di) implies ψ(dic). However, Xi ⊆ φ(x, y), so there cannot bemore than r elements d satisfying ψ(y, c), a contradiction.

Thus, RM(a/A) ≥ α + 1, and the proof is complete.

Theorem 4.1.14. Let D be a strongly minimal saturated structure, A ⊆ D and a ∈ D.Then dim(a/A) = RM(a/A) and dM(a/A) = mult(a/A).

Proof. We will show by induction on k that if a1, . . . , ak are independent over A, thenRM((a1, . . . , ak)/A) = k. For k = 1, let a1 6∈ acl(A) and take φ1 to be the formulain tp(a1/A) with RM(a1/A) = RM(φ1). Clearly, φ1(D) is infinite, and so we haveRM(φ1(D)) ≥ 1. Now, sinceD is strongly minimal, there cannot infinite disjoint definablesubsets X1, X2 ⊆ D. In particular, RM(φ1(D)) 6≥ 2, and we conclude that RM(a1/A) =RM(φ1) = 1.

Suppose now that a1, . . . , ak+1 are independent over A, and let φk+1(x1, . . . , xk, xk+1)and ψk+1(xk+1; a1, . . . , ak) ∈ tp(ak+1/Aa1 . . . ak) be formulas satisfying RM(φk+1) = RM(a1, . . . , ak+1

and RM(ψk+1) = RM(ak+1/Aa1 . . . ak).By the case k = 1, ψk+1 is infinite. Let bi : i < ω be different elements realizing

tp(ak+1/Aa1, . . . , ak). Now, let φk(x1, . . . , xk) ∈ tp(a1, . . . , ak/A) be a formula witnessingthat RM(φk) = RM(a1, . . . , ak/A) = k, by induction hypothesis. Then, since Mor-ley rank is invariant under definable bijections, we have that the formulas defined byθi(x1, . . . , xk+1) := “φk(x1, . . . , xk) ∧ xk+1 = bi” have all Morley rank k.

Also, since there is a unique type over A of any algebraically independent k+1-tuple,we have that

tp(a1, . . . , ak, ak+1/A) = tp(a1, . . . , ak, bi/A).

This implies that θi ⊆ φk+1 for every i < ω, and since they are disjoint we conclude thatRM(a1, . . . , ak+1/A) = RM(φk+1) ≥ k + 1.

On the other hand, since dM(Dk+1) = 1, there are no disjoint subsets of Dk+1 withrank ≥ k + 1. This completes the proof that RM(a1, . . . , ak+1/A) = k + 1.

We have shown the result for independent tuples of elements. Suppose now that a ∈ Dn

is an arbitrary tuple, and assume without loss of generality that a1, . . . , ak are indepen-dent and ak+1, . . . , an ∈ acl(A, a1, . . . , ak). Then, dim(a/A) = k = dim(a1, . . . , ak/A) =


RM(a1, . . . , ak/A) and by Lemma 4.1.13 we have

dim(a/A) = k = dim(a1, . . . , ak/A)

= RM(a1, . . . , ak/A) = RM(a1, . . . , ak, ak+1/A)

= · · · = RM(a/A).

Lemma 4.1.15. Let X1, . . . , Xn be definable sets. Then RM (⋃ni=1Xi) = maxRM(Xi) :

1 ≤ i ≤ n

Proof. First not that X ⊆ Y implies RM(X) ≤ RM(Y ) since any collection of non-emptydisjoint subsets of X can be taken to be subsets of Y too. By induction on ordinals, weprove that if RM (

⋃ni=1Xi) ≥ α for subsets X1, . . . , Xn of Dm then RM(Xi) ≥ α for some

i ≤ n.The case α = 0 is clear: if the union is not empty, then one of the sets must be also

non-emtpy. If α is a limit ordinal, it follows from induction hypothesis and the fact that,since the union is finite, one of the sets Xi must be mentioned infinitely often and thusmust have Morley rank bigger than β for all β < α.

Suppose now that RM (⋃ni=1Xi) ≥ α + 1, and let (Aj : j < ω) be a collection of

disjoint subsets of⋃ni=1Xi with RM(Aj) ≥ α for all j < ω.

Therefore, for each j < ω, we have Aj =⋃ni=1(Aj ∩Xi), and by induction hypothesis

there is an index ij ∈ 1, . . . , n such that RM(Aj ∩Xi) ≥ α. By piggeonhole principle,there is i ≤ n such that i = ij for infinitely many j, and (Xi ∩ Aj : ij = i) would bean infinite collection of pairwise disjoint subsets of Xi of Morley rank at least α. Thus,RM(Xi) ≥ α+ 1.

Remark 4.1.16. Note that if X ⊆ Dn, then dim(X) = maxdim(b/A) : b ∈ X =maxRM(b/A) : b ∈ X = RM(X), and clearly dM(X) = mult(X).

Fact 4.1.17. Suppose X ⊆ Dn and Y ⊆ Dm are non-empty definable sets. Then,

1. RM(X × Y ) = RM(X) +RM(Y ).

2. dM(X × Y ) = dM(X) · dM(Y ).

In particular, if dM(X) = dM(Y ) = 1 we also have that dM(X × Y ) = 1.

Proof.

1. We first show by induction on α that if RM(Y ) ≥ α then RM(X×Y ) ≥ RM(X)+α.

• For α = 0: If RM(Y ) ≥ 0, then Y 6= ∅, and sinceX is also non-empty we wouldhave, for a fixed y ∈ Y , RM(X×Y ) ≥ RM(X×y) = RM(X) = RM(X)+0.

• For α limit ordinal: If RM(Y ) ≥ α then RM(Y ) ≥ β for all β < α, and byinduction hypothesis, we would have RM(X×Y ) ≥ RM(X)+β for all β < α.Thus, RM(X × Y ) ≥ RM(X) + α.


• Suppose RM(Y ) ≥ α+1, and let (Yi : i < ω) be a collection of disjoint subsetsof Y with RM(Yi) ≥ α. Then, RM(X × Yi) ≥ RM(X) + α by inductionhypothesis, and since the sets (X × Yi : i < ω) are all disjoint, we haveRM(X × Y ) ≥ RM(X) + α + 1.

So, we have proved that RM(X × Y ) ≥ RM(X) +RM(Y ).

Notice that the inequality RM(X × Y ) ≥ RM(X) + RM(Y ) is not always true:if RM(X) = 1 and RM(Y ) = ω, then RM(X × Y ) = RM(Y × X) since thereis a definable bijection, but RM(X × Y ) ≥ RM(Y × X) ≥ ω + 1 > 1 + ω =RM(X) +RM(Y ).

However, if D is a saturated strongly minimal structure, the equality will followfrom the fact that dim(X × Y ) = dim(X) + dim(Y ). We have proved already thatdim(X × Y ) ≥ dim(X) + dim(Y ). Now we show that dim(X × Y ) ≤ dim(X) +dim(Y ).

Let X = X1 ∪ · · · ∪ Xr and Y = Y1 ∪ · · · ∪ Ys be decomposition of X and Yrespectively, with projections πXi : Dn → Ddim(X) such that πXi |Xi is finite-to-one, and respectively πYj for Y . Then, X × Y =

⋃1≤i≤r,1≤j≤sXi × Yj, and it

suffices to show that πij := πXi × πYj : Dn+m → Ddim(X)+dim(Y ) given by πij(x, y) =(πXi (x), π

Yj (y) is finite to one when restricted to Xi × Yj. But this is easy, since

π−1ij (c, d) = a ∈ Dn : πXi (a) = c × b ∈ Dm : πYj (b) = d is a cartesian product of

finite sets. Thus, since the projections πij |Xi×Yj are finite-to-one, we conclude thatdim(X × Y ) = dim(X) + dim(Y ) as desired.

2. If X = X1 ∪ · · · ∪ XdM(X) and Y = Y1 ∪ · · · ∪ YdM(Y ) be maximal decompositionsfor X and Y into disjoint sets of maximal rank. Then, X × Y =

⋃i,jXi × Yj

is a decomposition of X × Y into dM(X) · dM(Y ) sets of rank RM(Xi × Yj) =RM(Xi) + RM(Yj) = RM(X) + RM(Y ) = RM(X × Y ). Thus, dM(X × Y ) ≥dM(X) + dM(Y ).

Definition 4.1.18. Let B ⊆ M . We consider the transcendence dimension of B over Ato be the cardinality of a maximal subset of B independent over A. We denote this bydimtr(B/A).

Theorem 4.1.19. Suppose T is a strongly minimal theory. If M,N |= T then M ∼= N ifand only if dimtr(M) = dimtr(N).

More generally, if φ(x, a) is a strongly minimal formula with parameters from A withA ⊆ M,N , and dimtr(φ(M)) = dimtr(φ(N)) then there is a bijective partial elementarymap f : φ(M) → φ(N).

Proof. Let B be a transcendence basis for φ(M) and C be a transcendence basis for φ(N)over A (i.e., maximal subsets of φ(M) and φ(N) independent over A).

Since |B| = dimtr(φ(M)) = dimtr(φ(N)) = |C|, we can find a bijection f : B →C, and such bijection is actually A-elementary: if M |= φ(b1, . . . , bn) then φ has theform ¬ψ(x1, . . . , xn) for some formula ψ(x1, . . . , xn) with dimension < n. Given that


f(b1), . . . , f(bn) are different elements in the transcendence basis C ,we haveN |= ¬ψ(f(b1), . . . , f(bn))and so N |= φ(f(b1), . . . , f(bn)).

Now, let I be the collection of all partial A-elementary maps g : B′ → C ′ with f ⊆ g,B ⊆ B′ ⊆ φ(M), C ⊆ C ′ ⊆ φ(N). It is easy to show by Zorn’s Lemma that there isan element g ∈ I which is maximal under inclusion. We have g : B′ → C ′ is a partialA-elementary map, and now we show that B′ = φ(M) and C ′ = φ(N), completing theproof.

If b ∈ φ(M)\B′, then there is an algebraic formula ψ(x, d) isolating the type tpM(b/B′)(the formula in tp(b/B′) with the least possible number of solutions). Then M |=∃x(ψ(x, d)) for some d ⊆ B′, and so N |= ∃x(ψ(x, g(d))). Let c ∈ N be an elementsatisfying ψ(c, g(d)). By the minimality of ψ(x, d), we have tpM(b/B′) = tpN(c/C

′), andthe map g∪(b, c) would be a partial elementary map, contradicting the maximality of g.

Thus B′ = φ(M), and a similar argument shows that C ′ = φ(N), concluding thatg : φ(M) → φ(N) is a partial elementary map.

Corollary 4.1.20. If T is a countable strongly minimal theory, then T is κ-categoricalfor all κ ≥ ℵ1, and T has at most ℵ0 non-isomorphic models of size ℵ0 (also written asI(ℵ0, T ) ≤ ℵ0).

Proof. If M,N are two models of T of cardinality κ ≥ ℵ1, then since T is countable wehave dimtr(M) = ℵ1 = dimtr(N), and by the previous theorem, M ∼= N .

If M,N are countable models of T , M ∼= N iff dimtr(M) = dimtr(N), and so there areas many non-isomorphic models as possible values of the transcendence dimension, whichare countably many (namely, dimtr(M) ∈ N ∪ ℵ0). So, I(T,ℵ0) ≤ ℵ0.

Lemma 4.1.21. Suppose φ(x, y) is an L-formula and tp(a) = tp(b). Tjen RMD(φ(x, a)) =RMD(φ(x, b)) (that is, Morley rank is invariant under automorphisms).

Proof. By the symmetry of the statement, it is enough to show by induction on α that ifa ≡ b then RM(φ(x, a) ≥ α implies RM(φ(x, b)) ≥ α.

For α = 0: since a ≡ b, then D |= ∃x(φ(x, a)) ↔ ∃x(φ(x, b)), and so RM(φ(x, a)) ≥ 0if and only if RM(φ(x, b)).

For α limit ordinal, note that RM(φ(x, a)) ≥ α iff RM(φ(x, a) ≥ β for all β < α iff(by induction hypothesis) RM(φ(x, b) ≥ β for all β < α iff RM(φ(x, b)) ≥ α.

Now, suppose, that RM(φ(x, a)) ≥ α+ 1. Then there are definable sets (ψi(D|x|; ci) :

i < ω) which are pairwise disjoint subsets of φ(D|x|, a), of Morley rank at least α. Bysaturation of D, there is a sequence 〈di : i < ω〉 such that for every n < ω,

tp(a, c1, . . . , cn) = tp(b, d1, . . . , dn),

and so we have the collection (ψi(D|x|; di) : i < ω) of pairwise disjoint definable sub-

sets of φ(x, b), and by induction hypothesis, RM(ψi(x, di)) = RM(ψi(x, c)). Therefore,RM(φ(x, b)) ≥ α + 1.

Corollary 4.1.22. Suppose that T is ω-stable, M |= T , X ⊆ Mn and Y ⊆ Mm aredefinable subsets and f : X → Y is a definable finite-to-one function from X onto Y .Then, RM(X) = RM(Y ).


Proof. Recall that by the previous results, RM(X) = maxRM(a/A) : a ∈ X where Ais a set of parameters over which X, Y, f are defined. Let a ∈ X, b ∈ Y and a′ ∈ X suchthat: RM(X) = RM(a/A), RM(Y ) = RM(b/A) and f(a′) = b. Since f is finite-to-one,a and f(a) are A-interalgebraic (and also a′ and f(a′) = b), so by Lemma 4.1.13 we have

RM(X) = RM(a/A) = RM(a, f(a)/A) = RM(f(a)/A)

≤ RM(Y ) = RM(b/A) = RM(b, a′/A) = RM(a′/A)

≤ RM(X).

Lemma 4.1.23. Let T be strongly minimal and φ(x, y) an L-formula. The set

Yφ,k := a : RM(φ(D|x|; a)) ≥ k

is definable for each k ≤ n.

Proof. By induction on n. For n = 1, notice that by strong minimality and saturationthere is a number N ∈ N such that |φ(D; a)| ≥ N implies φ(D, a) is infinite. Then,RM(φ(D; a)) ≥ 1 if and only if D |= ∃≥Nx(φ(x, a)), and so Yφ,1(y) = ∃≥Nx(φ(x, a)) andYφ,0(y) = ∃x(φ(x, y)).

Suppose the result for all s ≤ n. We now prove it for n+ 1 by induction on k.Consider a formula φ(x1, . . . , xn+1, y) and put Φ(x1, . . . , xn; y) = ∃xn+1(φ(x1, . . . , xn, y)).

Clearly, if RM(Φ(Dn; a)) ≥ k then RM(φ(Dn+1, a)) ≥ k, since any decomposition of Φinto disjoint sets provides a similar decomposition of φ(Dn+1, a) into disjoint sets.

By induction hypothesis, there is a formula YΦ,k(y) such that YΦ,k(a) if and only ifRM(Φ(Dn; a)) ≥ k.

Consider now the set

θa(Dn) = b ∈ Dn : φ(b, xn+1; a) is infinite.

Notice that θa(Dn) = b ∈ Dn : |φ(b, xn+1; a)| ≥ N for an appropriate N ∈ N, and we

can consider the formula

θ(x1, . . . , xn; y) := ∃≥Nxn+1(φ(x, xn+1, y)).

Let (b, c) ∈ φ(Dn+1, a) be an element such that RM(φ(x, xn+1; a)) = RM(b, c/a). Thenwe have:

RM(φ(x, xn+1; a)) = RM(b, c/a)

= dim(b, c/a) = dim(b/a) + dim(c/ba).

But b |= Φ(x, a) and dim(c/ba) depends only on θ(b, a), and Φ(x, a) ∧ θ(x, a) definesexactly those elements in the projection which have fibers of dimension 1, so we have

RM(φ(x, xn+1; a)) ≥ k ⇔ RM(φ(x; a)) ≥ k or RM(Φ(x; a) ∧ θ(x; a)) ≥ k − 1

⇔ D |= YΦ,k(a) ∨ YΦ∧θ,k−1(a).

as desired.

4.2. STRONGLY MINIMAL PSEUDOFINITE STRUCTURES 77

4.2 Strongly minimal pseudofinite structures

We now take a look to strongly minimal pseudofinite theories. Strongly minimal theoriesare the “nicest” stable theories in various senses, and are defined/characterized by the factthat any definable subset of the universe of a model of T being either finite or cofinite. Asit turns out, the behavior of strongly minimal pseudofinite structures combines effectivelythe main features of pseudofiniteness and strongly minimal structures, and allows us toprovide a good control of the dimension and size of definable sets, as we will see in thissection.

Theorem 4.2.1. Let D be a (saturated) pseudofinite strongly minimal structures, and letq ∈ N∗ be the pseudofinite cardinality of D (q = |D|). Then,

1. For any definable (with parameters) set X ⊆ Dn, there is a polynomial PX(x) ∈Z[x] with positive leading coefficient such that |X| = PX(q). Moreover, RM(X) =degree(PX).

2. For any L-formula φ(x, y) there is a finite number of polynomials P1, . . . , Pk ∈ Z[x]and L-formulas ψ1(y), . . . , ψk(y) such that:

(a) ψi(y) : i ≤ k is a partition of the y-space.

(b) For any b, |φ(D|x|; b)| = Pi(q) if and only if D |= ψi(b).

Proof. First we prove (1) by induction on RM(X). If RM(X) = 0, then X is finite andwe can simply put PX(x) = a0 = |X|.

Now, suppose RM(X) = n ≥ 1 where X ⊆ Dm for some m ≥ n. If the result is truefor cells, then by 4.1.5, there is a decomposition of X into disjoint cells X = Y1∪ · · · ∪Yk,and if by putting PX(x) =

∑ki=1 PYi(x) we would have

|X| =k∑

i=1

|Yi| =k∑

i=1

PYi(q) = PX(q).

So, let us assume that X is an n-cell. Then there is a projection π : Dm → Dn such thatπ(X) has Morley rank n, and π|X is r-to-1 for some r ∈ N. So, |X| = r · |π(X)|.

Also note that |π(X)| = |Dn| − |(Dn \ π(X))|, and since Dn has multiplicity 1, wehave RM(Dn \ π(X)) < n. By induction hypothesis, there is a polynomial p ∈ Z[x] suchthat |Dn \ π(X)| = p(q). Then, we have

|X| = r|π(X)| = r(|Dn| − |(Dn \ π(X))|)= r(qn − p(q)) = rqn − r · p(q)

So, we can put PX(x) = r·xn−r·p(x). Also note that degree(p(x)) = RM(Dn\π(X)) < n,so we have that degree(PX(x)) = n = RM(X).

We now will prove (2). We start with the following claim.


Claim: For any L-formula φ(x, y) where x = (x1, . . . , xn) and a with |y| = |a|, there isan L-formula ψ(y) ∈ tp(a) such that for all b |= ψ(y) we have Pφ(x,a)(q) = Pφ(x,b)(q).

Proof of the Claim: By induction on n = |x|. For n = 1, by strong minimality, φ(x, a)defines a set that is either finite or cofinite. If D = φ(D; a)| = ℓ < ω, then Pφ(x,a)(q) = ℓand we can take ψφ(a) := ∃=ℓx(φ(x; a)). On the other hand, |¬φ(D, a)| = q − ℓ, and wecan take ψφ(a) = ∃=ℓx(¬φ(x; a)).

Suppose now the result for all formulas in at most n variables, and take a formulaφ(x1, . . . , xn, xn+1; y). As in previous proofs, consider the formula

Φ(x1, . . . , xn; y) := ∃xn+1(φ(x1, . . . , xn, xn+1; y).

By definability of Morley rankwe have φ(x; a) ≡ V0(x; a)∪V1(x; a) where Vs(x; a) =b ∈ Dn : dim(φ(b, xn+1; a)) = s for s = 0, 1.

Furthermore, by strong minimality we can write Vs(x; a) =⋃jWs,j(x, a) where we

have:

b ∈ W0,j(x; a) ⇔ V0(b; a) ∧ |φ(x; a)| = j

b ∈ W1,j(x; a) ⇔ V1(b; a) ∧ |¬φ(x; a)| = j.

By induction hypothesis, there exists formulas ψs,j(y) such that D |= ψs,j(a), and for allc ∈ D|y|,

D |= ψs,j(c) ⇔ |Ws,j(x; a)| = |Ws,j(x; c)|Note that j varies in a finite set of natural numbers by strong minimality and saturation,so we can now take the formula

ψφ(y) :=

(φ(x, y) ↔

∨

s,j

Ws,j(y)

)

∧(∀x (Ws,j(x, y) ↔ [dim(φ(x, xn+1, y)) = s ∧ |(φ(x, xn+1, y))

s| = j]) ∧∧

s,j

ψs,j(y)

)

By construction, D |= ψφ(a). Now, if c |= ψφ(y), then |= ∧s,j ψs,j(y) and we have

|φ(x, c)| =∑

j

|W0,j(x, c)| · j +∑

j

|W1,j(x, c)| · (q − j)

=∑

j

|W0(x, a)| · j +∑

j

|W1,j(x, a)| · (q − j)

= |φ(x, a)| ,

as we desired. Claim.

Note now that from the proof of the Claim that we may suppose, by induction hy-pothesis, that for each s, j there are finitely many formulas ψis,j(y) such that ψis,j(y)i is

a partition of the y-space, and for any b, |W is,j(x, b)| = Ps,j,i(q) if and only if |= ψis,j(b).

By varying along all possible combinations of indices s, j, i, we can conclude (2).

4.2. STRONGLY MINIMAL PSEUDOFINITE STRUCTURES 79

Corollary 4.2.2. We also have that dM(X) = ℓc(PX(q)), the leading coefficient of PX(q).

Proof. Suppose X ⊆ Dn, and let us prove the result by induction on n. For n = 1, bystrong minimality, X ⊆ D1 is either finite or cofinite. If X is cofinite, then PX(q) = q− c,and since X ⊆ D1 has maximal dimension dM(X) = 1 because mult(X) = 1. If X isfinite, then |X| = k for some k, and clearly dM(X) = k (there are k points).

Suppose now the result for all definable sets in at most n variables, and let X =φ(x1, . . . , xn+1; a) ⊆ Dn+1 and consider Φ(x1, . . . , xn; a) as in the proof of Theorem 4.2.1.By realizing the same decomposition we have

|X| = |φ(x1, . . . , xn+1; a)|=∑

j

j · |W0,j(x; a)|+∑

j

(q − j) · |W1,j(x; a)|

Suppose for the moment that X has Morley degree 1. By taking generic points there is atuple (b1, . . . , bn+1) ∈ X such that

RM(X) = dim((b1, . . . , bn+1)/a) = dim((b1, . . . , bn)/a) + dim(bn+1/ab1, . . . , bn).

Let s, j be indices such that b = (b1, . . . , bn) |= Ws,j(x; a). Then dim(bn+1(ab) = s,and also dim(b1, . . . , bn/a) ≤ Ws,j because Ws,j(x, a) is definable over a. So, dim(X) =dim(Ws,j) + s.

Clearly the set X ′ := φ(x1, . . . , xn+1; a) ∧ Ws,j(x1, . . . , xn; a) is contained in X and(b1, . . . , bn+1) ∈ X ′, so we have

dim(X) = dim(b1, . . . , bn+1/a) ≤ dim(X ′) ≤ dim(X) ⇒ dim(X ′) = dim(X).

Since we have assumed that the Morley degree of X is 1, we have dim(X \ X ′) <dim(X). We then have PX(q) = |X| = |X ′| + |X \ X ′| = PX′(q) + PX\X′(q) and sincedegree(PX\X′(q)) = dim(X \X ′) < dim(X) = degree(PX(q)), we have that ℓc(PX(q)) =ℓc(PX′(q)).

By a straightforward calculation we have

PX′(q) = |X ′| =|W0,j(x; a)| · j if s = 0

|W1,j(x, a)| · (q − j)if s = 1=

j · PW0,j

(q) if s = 0

(q − j) · PW1,j(q) if s = 1

In both cases, note that by induction hypothesis we have ℓc(PX(q)) = ℓc(PWs,j(q)) =(∗)

dM(Ws,j). It remains to show that dM(Ws,j) = 1.If s = 0, then dim(X) = dim(Ws,j). Also note that i Ws,j = Y1 ∪ Y2 for disjoint

sets Y1, Y2 of dimension dim(X), then the sets π−1X (Y1), π

−1X (Y2) would be disjoint, and

since π|X is a finite-to-one map, we have dim(π−1X (Yt)) = dim(Yt) = dim(X) for t = 1, 2,

contradicting that dM(X) = 1.Therefore, dM(Ws,j) = 1. Also note in this case we have j = 1 because otherwise we

could write X as the union of j-sets of dimension dim(X), by (*).Now, if s = 1, as in the previous case, if Y1∪Y2 = W1,j for disjoint sets of dimension

equal to dim(W1,j), then by additivity we would have

dim(π−1X (Yt) = dim(Yt) + 1 = dim(W1,j) + 1 = dim(X),

contradicting that dM(X) = 1.


4.2.1 Unimoduarity

Definition 4.2.3.

1. We say that two tuples a, b are interalgebraic if both tp(a/b) and tp(b/a) have afinite number of realizations.

2. When tp(a/b) we define the multiplicity of a over b to be the number of realizationsof tp(a/b).

Definition 4.2.4. A strongly minimal theory is said to be unimodular if whenever a =(a1, . . . , an) and b = (b1, . . . , bn) are algebraic independent n-tuples, and a is interalgebraicwith b, then

mult(a/b) = mult(b/a).

This notion is due to Hrushovski in [13], but a clarifications of the definition appearsin [18] and [9]. The most important result in [13] is the following:

Theorem 4.2.5 (Hrushovski, 1992). Unimodular strongly minimal sets are locally mod-ular.

This result generalizes Zilber’s result that ω-categorical strongly minimal sets arelocally modular.

Corollary 4.2.6. A strongly minimal pseudofinite structure D is unimodular. Hence, byTheorem 4.2.5, also locally modular.

Proof. Suppose a, b are n-tuples from D which are algebraicalally independent tuples, buta is interalgebraic with b. Write k = mult(b/a) and ℓ = mult(b/a). We have to prove thatk = ℓ.

Let φ(x, y) be an L-formula such that D |= φ(a, b), φ(x, b) isolates the type tp(a/b),and φ(a, y) isolates the type tp(b/a). (It can be the conjuction of the formulas thatisolate each of the types). Let φ1(x) := ∃=ky(φ(x, y)) and φ2(y) := ∃=ℓx(φ(x, y)). Letχ(x, y) := φ(x, y) ∧ φ1(x) ∧ φ2(y), and take Z ⊆ D2n the set defined by χ(x, y), and putX = πx(Z) (the projection of Z on the first n coordinates) and Y = πy(Z) (the projectionof Z on the last n coordinates).

Then, by counting on the finite structures, |Z| = k · |X| = ℓ|Y |, and this implies byTheorem 4.2.1 that k · PX(q) = ℓ · PX(q). (∗)

On the other hand, since a ∈ X ⊆ Dn and b ∈ Y ⊆ Dn, we have RM(X) ≤ n,RM(Y )and also

RM(X) ≥ RM(a) = dim(a) = n and RM(Y ) ≥ RM(b) = dim(b) = n.

So RM(X) = n = RM(Y ), and since mult(Dn) = 1, we conclude that dM(X) =dM(Y ) = 1. This implies that the leading term of PX(q) and PY (q) is qn, so by comparingthe leading terms in both sides of the equality (∗) we get k · qn = ℓ · qn, which impliesk = ℓ.

Bibliography

[1] V. Bergelson, A. Blass, M. Di Nasso, R. Jin, eds. Ultrafilters across mathematics.Contemporary Mathematics, 530. 2010

[2] V. Bergelson. Ergodic theory and Diophantine problems. In Topics in symbolic dy-namics and applications (Temuco, 1997), volume 279 of London Math. Soc. LectureNote Ser., pages 167–205. Cambridge Univ. Press, Cambridge, 2000.

[3] B. Bollobás. Random graphs. Academic Press. New York, 1985.

[4] G. Cherlin, L. Harrington, A.H. Lachlan. ℵ0-categorical, ℵ0-stable structures. Ann.Pure Appl. Logic, 28(2):103–135, 1985.

[5] A. Chernikov, I. Kaplan. Forking and dividing in NTP2 theories. J. Symbolic Logic.Voume 77, Issue 1 (2012), 1-20.

[6] A. Chernikov, S. Starchenko. A note on the Erdös-Hajnal property for stable graphs.Available at http://arxiv.org/pdf/1504.08252v2.pdf . 2016

[7] H. Furstenberg. Ergodic behavior of diagonal measures and a theorem of Szemeredion arithmetic progressions. Journal d’Analyse Mathématique, 31:204–256, 1977.10.1007/BF02813304.

[8] D. García, D. Macpherson, C. Steinhorn. Pseudofinite structures and simplicity. J.Math. Log. 15, 1550002 (2015)

[9] D. García, F. Wagner. Unimodularity unified. Journal of Symbolic Logic, Vol 60,2016.

[10] I. Goldbring, H. Towsner. An approximate logic for measures. Israel Journal of Math-ematics. March 2014, Volume 199, Issue 2, pp 867–913.

[11] Hajek. Generalized quantifiers and finite sets. 1977

[12] D. Haskell, A. Pillay, C. Steinhorn. Model Theory, Algebra and Geometry. Mathe-matical Sciences Research Institute Publications, 39. 2000.

[13] E. Hrushovski. Unimodular minimal sets. Journal of the London Math. Society, Vol-ume 46 no. 3 (1992), 385–396.

81

82 BIBLIOGRAPHY

[14] E. Hrushovski, F. Wagner. Counting and dimensions. Model Theory with applicationsto Algebra and Analysis. 2008

[15] E. Hrushovski. Stable group theory and approximate subgroups. Journal of the Amer-ican Mathematical Society, vol. 25 (2012), pp. 189–243

[16] E. Hrushovski. On pseudofinite dimensions. Notre Dame J. Formal Logic. Volume54, Number 3-4 (2013). 463-495

[17] J. Łoś, Quelques remarques, théorèmes problèmes sur les classes définissablesd’algèbres, pp. 98-113 in Mathematical Interpretations of Formal Systems, North-Holland 1955.

[18] C. Kestner, A. Pillay. Remarks on unimodularity. J. Symbolic Logic. Volume 76 no.4 (2011), 1453–1458.

[19] B. Kim, A. Pillay. Simple theories. Annals of Pure and Applied Logic, 88, pp. 149-164.1997.

[20] A. Kruckman. Disjoint n-amalgamation and pseudofinite countably categorical theo-ries. Available at http://arxiv.org/pdf/1510.03539v1.pdf . 2015.

[21] H.J. Keisler. Ultraproducts of finite sets. The Journal of Symbolic Logic Vol. 32, No.1 (Mar., 1967), pp. 47-57.

[22] A. Koponen, K. Luosto. Definability issues in finite group theory. Reports of theDepartment of Mathematics, University of Helsinki, 227. 1999

[23] M. Malliaris, S. Shelah. Regularity lemmas for stable graphs. Trans. Amer. Math.Soc. 366 (2014), 1551-1585.

[24] A. Pillay. Strongly minimal pseudofinite structures. Available at http://arxiv.org/pdf/1411.5008v2.pdf . 2014

[25] S. Shelah. On the cardinality of ultraproducts of finite sets. J. Symbolic Logic 35(1970), no. 1, 83–84

[26] T. Tao. Expanding polynomials over finite fields of large characteristic, and a regu-larity lemma for definable sets. Contributions to Discrete Mathematics. Volume 10,Number 1, Pages 22–98, ISSN 1715-0868. 2014

[27] T. Tao. A correspondence principle between (hyper) graph theory and probability the-ory, and the (hyper) graph removal lemma. Journal d’Analyse Mathématique. De-cember 2007, Volume 103, Issue 1, pp 1–45

[28] G. Turán. On the definability of properties of finite graphs. Discrete Mathemat-ics,Volume 49, Issue 3, May 1984, Pages 291-302.

[29] F. Wagner. Simple theories. Kluwer academic publishers. 2000.

Documents

Model theory of pseudoﬁnite structures