Ramsey classes - math.wisc.edulempp/conf/asl12//Slides/Sokic.pdf · Introduction Motivation...

Ramseyclasses

Miodrag Sokic

IntroductionMotivationTopologicaldynamicsModel theoryRamsey theoryThe KPT theory

QuestionsPosetsUltrametricspaces

ResultsPosetsUltrametric

IntroductionQuestionsResults

Ramsey classes

Miodrag Sokic

Caltech

April 1st 2012, Madison

Miodrag Sokic Ramsey classes

Ramseyclasses

Miodrag Sokic

MotivationTopological dynamicsModel theoryRamsey theoryThe KPT theory

Motivation

topological dynamics

model theory

combinatorics

group actions on compact spaces

Fraïssé theory

Ramsey theory

Ramseyclasses

Miodrag Sokic

Motivation

model theory

combinatorics

Fraïssé theory

Ramsey theory

Ramseyclasses

Miodrag Sokic

Motivation

model theory

combinatorics

Fraïssé theory

Ramsey theory

Ramseyclasses

Miodrag Sokic

Motivation

model theory

combinatorics

Fraïssé theory

Ramsey theory

Ramseyclasses

Miodrag Sokic

Motivation

model theory

combinatorics

Fraïssé theory

Ramsey theory

Ramseyclasses

Miodrag Sokic

Motivation

model theory

combinatorics

Fraïssé theory

Ramsey theory

Ramseyclasses

Miodrag Sokic

Motivation

model theory

combinatorics

Fraïssé theory

Ramsey theory

Ramseyclasses

Miodrag Sokic

Topological dynamics

G-�ow continuous action on a compact

Hausdor¤ space

G� X ! X

minimal �ow for all x 2 X

G � x = X

FactEvery G-�ow contains a minimal sub�ow.

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Miodrag Sokic

Hausdor¤ space

G� X ! X

G � x = X

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Miodrag Sokic

Hausdor¤ space

G� X ! X

G � x = X

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Miodrag Sokic

Theorem

Given a topological group G, there is a minimal G-�ow M(G)such that for any other minimal G-�ow Y there exists surjectivehomomorphism

ϕ : M(G)! Yϕ(gx) = gϕ(x)

Moreover such G-�ow M(G) is uniquely determined up toisomorphism.

Ramseyclasses

Miodrag Sokic

The minimal G-�ow M(G) established in the previous theoremis called the universal minimal G-�ow.

A group G is extremely amenable(has �xed point on compactaproperty) if M(G) = f�g.

Ramseyclasses

Miodrag Sokic

The minimal G-�ow M(G) established in the previous theoremis called the universal minimal G-�ow.A group G is extremely amenable(has �xed point on compactaproperty) if M(G) = f�g.

Ramseyclasses

Miodrag Sokic

Fraïssé classes

K-class of �nite structures in a countable signature L, closedunder isomorphic imagesFraïssé class

contains structures of arbitrary large �nite cardinality

hereditary property (HP) A ,! B 2 K =) A 2 Kjoint embedding property (JEP) (8A, B 2 K)(9C 2 K)[A ,! C & B ,! C]amalgamation property (AP) (8A, B, C 2 K)(8f : A! B, g : A! C)(9D 2 K)(9f 0 : B! D, g0 : C! D)[ f 0 � f = g0 � g]

Ramseyclasses

Miodrag Sokic

Fraïssé classes

hereditary property (HP) A ,! B 2 K =) A 2 K

joint embedding property (JEP) (8A, B 2 K)(9C 2 K)[A ,! C & B ,! C]amalgamation property (AP) (8A, B, C 2 K)(8f : A! B, g : A! C)(9D 2 K)(9f 0 : B! D, g0 : C! D)[ f 0 � f = g0 � g]

Ramseyclasses

Miodrag Sokic

Fraïssé classes

hereditary property (HP) A ,! B 2 K =) A 2 Kjoint embedding property (JEP) (8A, B 2 K)(9C 2 K)[A ,! C & B ,! C]

amalgamation property (AP) (8A, B, C 2 K)(8f : A! B, g : A! C)(9D 2 K)(9f 0 : B! D, g0 : C! D)[ f 0 � f = g0 � g]

Ramseyclasses

Miodrag Sokic

Fraïssé classes

hereditary property (HP) A ,! B 2 K =) A 2 Kjoint embedding property (JEP) (8A, B 2 K)(9C 2 K)[A ,! C & B ,! C]amalgamation property (AP) (8A, B, C 2 K)(8f : A! B, g : A! C)(9D 2 K)(9f 0 : B! D, g0 : C! D)[ f 0 � f = g0 � g]

Ramseyclasses

Miodrag Sokic

Fraïssé structures

Fraïssé structure (countable signature L)

in�nite countable

locally �nite-every �nitely generated substructure is �nite

ultrahomogeneous-any isomorphism between two �nitesubstructures can be extended to (global) automorphism

Age(A)-(age)class of �nite structures embedable into A

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Miodrag Sokic

in�nite countable

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Miodrag Sokic

in�nite countable

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Miodrag Sokic

Theorem

Let K be a Fraïssé class. Then there is a unique, up toisomorphism, Fraïssé structure K such that K = Age(K).

Fraïssé limit-K =F lim(K)

Theorem

Let K be a Fraïssé structure. Then Age(K) is a Fraïssé class.

For every Fraïssé structure K we consider its automorphismgroup Aut(K) with pointwise convergence topology,it is aPolish group.

Ramseyclasses

Miodrag Sokic

Theorem

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Miodrag Sokic

Theorem

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Miodrag Sokic

Arrow notation

K-class of �nite structures in signature L

For A, B 2K we write:�BA

�= fC � B : C u Ag.

Let A, B, C 2K. If for any coloringc : (C

A)! f1, .., rg,there exist a structure B0 2 (C

B) such that

c ��

�= const

then we writeC! (B)A

Ramseyclasses

Miodrag Sokic

Arrow notation

K-class of �nite structures in signature LFor A, B 2K we write:�

�= fC � B : C u Ag.

B) such that

c ��

�= const

Ramseyclasses

Miodrag Sokic

Arrow notation

K-class of �nite structures in signature LFor A, B 2K we write:�

�= fC � B : C u Ag.

B) such that

c ��

�= const

Ramseyclasses

Miodrag Sokic

Ramsey degree

Let K be a class of �nite structures. If for natural numbers rand t and structures A, B and C from K we have that forevery coloring

c :�

�! f1, ..., rg,

there is B0 2 (CB) such that c � (B0

A) takes at most t many

values then we writeC! (B)Ar,t.

We say that A 2 K has the Ramsey degree t if for everyB 2K and every natural number r there is C 2K such thatC! (B)Ar,t and t is the smallest such number. We denote sucha number by tK(A).Note that K is a Ramsey class i¤ tK(A) = 1 for all A 2K.

Ramseyclasses

Miodrag Sokic

(Kechris-Pestov-Todorµcevic 2005)

TheoremFor ordered classes

RP ! extreme amenability,(RP for K) ! Aut(F lim(K)).

Theorem

(RP+OP) ! universal minimal �ow,(RP for K+OP for OK) !

universal minimal �ow Aut(F lim(K)).

Ramseyclasses

Miodrag Sokic

(Kechris-Pestov-Todorµcevic 2005)

TheoremFor ordered classes

RP ! extreme amenability,(RP for K) ! Aut(F lim(K)).

Theorem

(RP+OP) ! universal minimal �ow,(RP for K+OP for OK) !

universal minimal �ow Aut(F lim(K)).

Ramseyclasses

Miodrag Sokic

PosetsUltrametric spaces

Posets

Let P be the class of �nite posets.If we add arbitrary linear orderings to posets:

OP = f(A,v,�) : (A,v) 2 P & �2 lo(A)g.

If we add only linear extensions of partial orderings:

EP = f(A,v,�) 2 OP :�2 le(v)g.

Ramseyclasses

Miodrag Sokic

Limits

Fraïssé limits of posets:

Random poset P = F lim(P),Random poset with linear extension EP = F lim(EP),Random poset with random ordering OP = F lim(OP).

Ramseyclasses

Miodrag Sokic

Problems

UMF for Aut(P) isknown.

Aut(EP) is extremelyamenable.

Ramsey degree forelements in P is known.EP is a Ramsey class.OP is not a Ramseyclass.

Calculate UMF forAut(OP).

Calculate Ramsey degreesfor objects in OP .

Ramseyclasses

Miodrag Sokic

Problems

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Miodrag Sokic

Problems

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Miodrag Sokic

Ultrametric

A metric d on a set X is an ultrametric if for all x, y, z 2 X wehave

d(x, z) � maxfd(x, y), d(y, z)g.The spectrum of a metric space X = (X, d) is the set

Spec(X) = fd(x, y) : x 2 X, y 2 X, x 6= yg.

We denote the class of �nite ultrametric spaces by U , and forS � (0,+∞), we consider the class

US = fX 2U : Spec(X) � Sg.

Ramseyclasses

Miodrag Sokic

Ordered spaces

Linearly ordered ultrametric spaces with distances in the set S :

OU S = f(X, d,� ) : (X, d) 2 US & �2 lo(X)g.

If (X, d,� ) 2OU S and for all x, y, z 2 X we have

x � y � z =) d(x, y) � d(x, z),

then we say that the linear ordering � is convex on theultrametric space (X, d).In the convex linear ordering every ball is an interval withrespect to the linear ordering.The class of all convex linear orderings of an ultrametric spaceX = (X, d) we denote by co(X), and we consider:

CU S = f(X, d,� ) 2OU S : �2 co((X, d))g

Ramseyclasses

Miodrag Sokic

Ordered spaces

OU S = f(X, d,� ) : (X, d) 2 US & �2 lo(X)g.

x � y � z =) d(x, y) � d(x, z),

then we say that the linear ordering � is convex on theultrametric space (X, d).

In the convex linear ordering every ball is an interval withrespect to the linear ordering.The class of all convex linear orderings of an ultrametric spaceX = (X, d) we denote by co(X), and we consider:

CU S = f(X, d,� ) 2OU S : �2 co((X, d))g

Ramseyclasses

Miodrag Sokic

Ordered spaces

OU S = f(X, d,� ) : (X, d) 2 US & �2 lo(X)g.

x � y � z =) d(x, y) � d(x, z),

then we say that the linear ordering � is convex on theultrametric space (X, d).In the convex linear ordering every ball is an interval withrespect to the linear ordering.

The class of all convex linear orderings of an ultrametric spaceX = (X, d) we denote by co(X), and we consider:

CU S = f(X, d,� ) 2OU S : �2 co((X, d))g

Ramseyclasses

Miodrag Sokic

Ordered spaces

OU S = f(X, d,� ) : (X, d) 2 US & �2 lo(X)g.

x � y � z =) d(x, y) � d(x, z),

then we say that the linear ordering � is convex on theultrametric space (X, d).In the convex linear ordering every ball is an interval withrespect to the linear ordering.The class of all convex linear orderings of an ultrametric spaceX = (X, d) we denote by co(X), and we consider:

CU S = f(X, d,� ) 2OU S : �2 co((X, d))g

Ramseyclasses

Miodrag Sokic

Limits

Fraïssé limits of ultrametric spaces when S is countable:

Random ultrametric space with spectrumS:US = F lim(US).

Random convexly ordered ultrametric space with spectrumS: CUS = F lim(CU S).

Random ordered ultrametric space with spectrum S:OUS = F lim(OU S).

Ramseyclasses

Miodrag Sokic

Problems

UMF for Aut(US) isknown.

Aut(CUS) is extremelyamenable.

Ramsey degree forelements in US is known.

CU S is a Ramsey class.

OU S is not a Ramseyclass.

Calculate UMF forAut(OUS).

Calculate Ramsey degreesfor objects in OU S.

Ramseyclasses

Miodrag Sokic

Problems

Ramseyclasses

Miodrag Sokic

Problems

Ramseyclasses

Miodrag Sokic

PosetsUltrametric

Expansion

We consider posets with two linear orderings:

EOP = f(A,v,�,�) : (A,v,�) 2 OP & (A,v,�) 2 EPg.

Ramseyclasses

Miodrag Sokic

PosetsUltrametric

Theorem

EOP is a Ramsey class which satis�es OP with respect to OP .

Corollary

tOP (A) = jle(v)j.

Corollary

UMF of Aut(OP) is a closed subset of the compact space ofall linear orderings LO � 2N�N.

Ramseyclasses

Miodrag Sokic

PosetsUltrametric

Theorem

Corollary

tOP (A) = jle(v)j.

Corollary

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Miodrag Sokic

PosetsUltrametric

Theorem

Corollary

tOP (A) = jle(v)j.

Corollary

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Miodrag Sokic

PosetsUltrametric

Theorem

Corollary

tOP (A) = jle(v)j.

Corollary

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Miodrag Sokic

PosetsUltrametric

Expansions

Ultrametric spaces with two orderings:

COU S = f(X, d,�,� ) : (X, d,�) 2 OU S & (X, d,�) 2 CU Sg.

In the case that the set S � (0,+∞) has a minimumv = min(S), we consider the class:

COUminS = f(X, d,�,� ) 2COU S : (8x 2 X) �� Bv[x] =�� Bv[x]g.

Ramseyclasses

Miodrag Sokic

PosetsUltrametric

Expansions

Ramseyclasses

Miodrag Sokic

PosetsUltrametric

Expansions

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Miodrag Sokic

PosetsUltrametric

Theorem

For non empty S � (0,+∞), COU S is a Ramsey class.

For non empty S � (0,+∞) where S has a minimum,COUmin

S is a Ramsey class.

Theorem

Let 0 /2 S � (0,+∞) be such that inf(S) /2 S. Then theclass COU S satis�es the ordering property with respect toOU S.

Let 0 /2 S � (0,+∞) be such that inf(S) 2 S. Then theclass COUmin

S satis�es the ordering property with respectto OU S.

Ramseyclasses

Miodrag Sokic

PosetsUltrametric

Theorem

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Miodrag Sokic

PosetsUltrametric

Theorem

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Miodrag Sokic

PosetsUltrametric

Theorem

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Miodrag Sokic

PosetsUltrametric

Theorem

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Miodrag Sokic

PosetsUltrametric

Theorem

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Miodrag Sokic

PosetsUltrametric

Theorem

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Miodrag Sokic

PosetsUltrametric

Application

Theorem

Let S � (0,+∞) be non empty without a minimalelement. Then for A = (A, d,�)2OU S we have

tOUS(A) = jco((A, d))j.

Let S � (0,+∞) be non empty with a minimal element.Then for A 2OU S we have

tOUS(A) = jf�: (A, d,�,�) 2 COUminS gj.

Corollary

UMF of Aut(OUS) is closed subset of LO.

Ramseyclasses

Miodrag Sokic

PosetsUltrametric

Application

Theorem

Corollary

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Miodrag Sokic

PosetsUltrametric

Application

Theorem

Corollary

Ramseyclasses

Miodrag Sokic

PosetsUltrametric

Application

Theorem

Corollary

Ramseyclasses

Miodrag Sokic

PosetsUltrametric

Application

Theorem

Corollary

Ramseyclasses

Miodrag Sokic

PosetsUltrametric

Thank you

Ramsey classes - math.wisc.edulempp/conf/asl12//Slides/Sokic.pdf · Introduction Motivation...

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