Pierre Matet and Saharon Shelah- Positive Partition Relations for P-kappa(lambda)

8/3/2019 Pierre Matet and Saharon Shelah- Positive Partition Relations for P-kappa(lambda)

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POSITIVE PARTITION RELATIONS FOR P()P()P()P()P()P()P()

PIERRE MATET and SAHARON SHELAH*

Abstract. Let a regular uncountable cardinal and a cardinal > , and suppose .> .> .> .> .

The paper is organized as follows. Section 1 reviews a number of standard definitions concerning idealson and P(). Sections 2-7 give results about combinatorics on that are needed for our study ofP(). Sections 2 and 3 review some facts concerning, respectively, the dominating number d and the

*This research was supported by the Israel Science Foundation. Publication 804.

2000 Mathematics Subject Classification : 03E02, 03E35, 03E55, 03E04Key words and phrases : P(), partition relation, weakly compact cardinal.

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covering number for category cov(MMMMMMM,). Section 4 deals with the problem of determining the value of theunequality number U in the case where is a successor cardinal. In Section 5 we show that if 2


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Let [A]2 = {(, ) A A : < } for any A . Given an ordinal 2, (, )2 means that for

every f : []22, there is A such that either A has order type and f is identically 0 on [A]2,

or A has order type and f is identically 1 on [A]2. The negation of this and other partition relationsis indicated by crossing the arrow. ()2 means that (, )2.

is weakly compact if ()2.

If is weakly compact, then it is inaccessible (see e.g. Proposition 4.4 in [Ka]).

An ideal J on is a weak P-point if given A J+ and f A with {f1({}) : < } J, thereis B J+ P(A) such that f is < -to-one on B. J is a local Q-point if given g , there isB J+ such that g() < for any (, ) [B]2. J is a weak Q-point if J | A is a local Q-point forevery A J+.

It is well-known (see [M1] for a proof) that an ideal J on is a weak Q-point if and only if given A J+

and a < -to-one f : A, there is B J+ P(A) such that f is one-to-one on B.

An ideal J on is weakly selective if it is both a weak P-point and a weak Q-point.

Given a cardinal with 2 and an ideal J on , J+[J+]2 means that for every A

J+ and every f : [A]2, there is B J+ P(A) such that f[B]2 = . []2 means that

(P())+[(P())+]2.

Note that []22 if and only if ()2.

Let P be a property such that at least one ideal on does not satisfy P. Then non(P) (respectively,non(P)) denotes the least cardinal for which one can find an ideal J on such that cof(J) =

(respectively, cof(J) = ) and J does not satisy P.

Notice that


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2. Domination

In this section we recall some characterizations of the dominating number d.

Definition. d is the least cardinality of any X with the property that for every g , there isf X such that g() < f() for all < .

d is the least cardinality of any X with the property that for every g , there is x P(X)such that g() ) = nonnonnonnonnonnonnon(weak P-point).

PROPOSITION 2.3. d nonnonnonnonnonnonnon(aJ > ) nonnonnonnonnonnonnon(weak P-point).

Proof. The first inequality follows from Proposition 2.1 (ii) since aNS = ([MP2]). To prove the secondinequality, argue as for Lemma 8.5 below.

QUESTION. Is it consistent that d > nonnonnonnonnonnonnon(weak P-point) ?

3. Covering for category

Throughout this section will denote a fixed regular infinite cardinal.

We will review some basic facts concerning the covering number cov(MMMMMMM,).

Definition. Suppose is a cardinal .Let F n(, 2, ) = {a2 : a P()}. F n(, 2, ) is ordered by : p q if and only if q p.2 is endowed with the topology obtained by taking as basic open sets and Os for s F n(, 2, ),where Os = {f

2 : s f}.MMMMMMM, is the set of all W 2 such that W (X) = for some collection X of dense open subsets of2 with 0


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PROPOSITION 3.2. Suppose that is a cardinal > and V |= 2 .

(iii) Let be any regular cardinal > . Then (d)VP

= (d)V and (d)

VP (d)V .

Proof. (iii) : The conclusion easily follows from the following observation : Suppose that is a cardinal> 0 and F VP is a function from to . Then by Lemma VII.6.8 of [K], there is H : P+()

such that H V and F(, ) H(, ) (so F(, ) H(, )) for every (, ) .

Remark. It is not known whether it is consistent that cf(cov(MMMMMMM,) .

4. Unequality

Our main concern in this section is with the problem of evaluating the unequality number U when is

a successor cardinal.

Definition. U (respectively, U) is the least cardinality of any F

with the property that for everyg , there is f F such that { : f() = g()} = (respectively | { : f() = g()} |< ).

The following is readily checked.

PROPOSITION 4.1. cov(MMMMMMM,) U d.

Remark. It is shown in [MRoS] that if V |= GCH, then there is a -complete +cc forcing notion Psuch that

VP |= d = + and cov(MMMMMMM,) = 2

= +(+1).

For models where d > + see also [CS].

PROPOSITION 4.2. U = U.

Proof. Fix F with the property that for every g , there is f F such that

| { : f() = g()} |< .

For f F and , < , define f, by : f,() = f() if , and f,() = otherwise. Then forevery g , there are f F and , < such that { : f,() = g()} = .

The following is due to Landver [L2].

PROPOSITION 4.3. cf(U) > .

Proof. Suppose otherwise. Set = cf(U) and fix F so that | F |= U and for every g , thereexists f F with { : f() = g()} = . Let < F : < > be such that (a) | F |< U for any , and

(b)


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We now turn our attention to the task of computing U. We begin with the case when is a successorcardinal.

THEOREM 4.4. Suppose is the successor of a regular infinite cardinal . Then

U min(d, cov(MMMMMMM,)).

Proof. Fix F with 0 f()} |=

for every f F. Select a bijection j : and a bijection i : k() for each < . Given

A and t A2, define a partial function t from to by stipulating that t() = if and only if(a) < k(), (b) {j(, ) : < i()} t

1({0}), and (c) j(, i()) t1({1}). For f F, let Df be

the set of all s F n(, 2, ) such that there is dom(s) with k() > f() and s() = f(). Clearly,

each Df is a dense subset of F n(, 2, ), so we can find g

2 with the property that for every f F,there is a P() with g a Df. Then

{ dom(g) : g() = f()} =

for every f F.

THEOREM 4.5. Suppose is a successor cardinal. Then U d.

Proof. Fix F with 0 by a result of Ulam (see [Ka], 16.3), there exist pairwise disjoint Di J+

for i < withi


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Now fix f F. Pick Dif C B{f} and set = gf(). Notice that by the definition of gf.Also, h() since B{f}. Hence c = by the definition of C and the fact that C. It nowfollows from the definition of t and the fact that Dif that (t()) = if. On the other hand, Afsince = gf(), so f() < k() and (f()) = if. Thus t() = f().

Remark. It follows from Proposition 4.1 and Theorem 4.5 that U = d if is a successor cardinal and

d < +.

THEOREM 4.6. Suppose that is a successor cardinal and 2


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Definition. An ideal J on is a weak semi-Q-point if given A J+ and a < -to-one function ffrom A to , there is C J+ P(A) such that | C f1({}) || | for every .J is weakly semiselective if J is both a weak semi-Q-point and a weak P-point.J is weakly rapid if given A J+ and f , there is C J+ P(A) such that o.t.(C f()) + 1for every .

Remark. It is simple to see that every weak Q-point ideal on is weakly rapid, and every weakly rapidideal on is a weak semi-Q-point.

Every weak semi-Q-point ideal on is weakly rapid ([MP1]). We will show that this does not generalize.

Definition. An ideal J on is a semi-Q-point if given a < -to-one function f from to , thereis B J such that | f1({}) B || | for every .

PROPOSITION 5.2. Suppose is a limit cardinal. Then there exists a semi-Q-point ideal on thatis not weakly rapid.

Proof. Let Y be the set of all infinite cardinals < . Select h Y

so that (a) h() is a regular infinitecardinal for every Y, and (b) { Y : h() } is stationary in for every Y. ForA and Y, let TA be the set of all Y such that h() and | A [, + h()) |= h().Now let Jh be the set of all A such that TA is a nonstationary subset of for some Y. It issimple to check that Jh is an ideal on .

Let us remark in passing that if is weakly Mahlo and h is defined by : h() = if is singular, andh() = otherwise, then a subset A of lies in Jh if and only if the set of all Y such that isregular and | A [, + ) |= is nonstationary in .

Let us show that Jh is a semi-Q-point. Thus fix a < -to-one function f : . Then

C =

Y : = . Then < andf1({}) . Thus

Q f1({}) [, + h())

and consequently

| Q f1({}) | h() | | .

It remains to show that J is not weakly rapid. Fix D J+h . Then

S = { TD :| TD |= }

is a stationary subset of . Given S, | D |= since

D [, + h()) D

for every TD , and henceo.t.(D ( + h()) > + 1.

THEOREM 5.3. Suppose is a limit cardinal. Then U nonnonnonnonnonnonnon(weak semi-Q-point).

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Proof. Let J be an ideal on with cof(J) < U. Let us show that J is a weak semi-Q-point. Thus fix

A J+ and a < -to-one function f : A. Select B J for < cof(J) so that J =

f1

({})) B.

There is h such that { : g() = h()} = for every < cof(J). Define C ran(h) by : h() C

just in case h() >

f1({}). Then clearly C J+ P(A). Moreover, C f1({}) {h() : < }

for every < .

THEOREM 5.4. Suppose that is a limit cardinal and 2


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Definition. Given an ideal J on , A J+ and F : 2, (J,A,F) is 0-good if there is

D J+ P(A) such that { D : F(, ) = 1} J for every D.


LEMMA 6.2. Suppose that J is weakly selective and (J,A,F) is 0-good, where J is an ideal on, A J+ and F : 2. Then there is B J+ P(A) such that F is constantly 0 on [B]2.

LEMMA 6.3. Suppose that (J,A,F) is not 0-good, where J is an ideal on , A J+ and F : 2.

Then :

(i) There is B A such that o.t.(B) = + 1 and F is identically 1 on [B]2.

(ii) Suppose that aJ > and is an uncountable cardinal < such that (, )2. Then there

is C A such that o.t.(C) = + 1 and F is identically 1 on [C]2.

Proof. The proof is similar to that of Lemma 10.4 below.

THEOREM 6.4.

(i) nonnonnonnonnonnonnon(J+(J+, + 1)2) nonnonnonnonnonnonnon(weakly selective).

(ii) Suppose that is an infinite cardinal < such that (, )2. Then

non(J+

(J+

, + 1)2

) non(weakly selective).

Proof. (i) : Baumgartner, Taylor and Wagon [BauTW] showed that J+(J+, + 1)2 for every weakly

selective ideal J on .

(ii) : By Lemmas 6.2 and 6.3.

Remark. Suppose that is a successor cardinal and is cardinal 2 such that (, )2. Then

by Theorems 6.1 (i), 6.4 (ii) and 2.2 and Proposition 5.1, d = nonnonnonnonnonnonnon(J+(J+, + 1)2).

Remark. It is consistent (see [M2]) that d > nonnonnonnonnonnonnon(J+(J+, 3)2). We do not know whether this can

be generalized.

THEOREM 6.5. Suppose is a weakly compact cardinal. Then :

(i) nonnonnonnonnonnonnon(weak Q-point) nonnonnonnonnonnonnon(J+(J+, )2).

(ii) nonnonnonnonnonnonnon(J+(J+)2) = nonnonnonnonnonnonnon(J+(J+, )2) = nonnonnonnonnonnonnon(weakly selective).

Proof. The result follows from Theorems 2.2 and 6.1 (i) and the following two well-known facts : (1) Everyideal J on such that J+(J+, )2 is a weak Q-point ; (2) If is weakly compact, then J+(J+)2

for every weakly selective ideal J on such that aJ > .

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7. nonnonnonnonnonnonnon(J+[J+]2)

In this section we consider the cardinal nonnonnonnonnonnonnon(J+[J+]2), where 3 , about which little is known.

We begin with the case where = 3. The following is due to Blass [Bl].

LEMMA 7.1. Suppose J is an ideal on such that J+[J+]23. Then J is a weak P-point.

Proof. Fix A J+ and f A with {f1({}) : } J. Define g : [A]23 by stipulating

that g(, ) = 0 if and only if f() < f(), and g(, ) = 1 if and only if f() = f(). There areB J+ P(A) and i < 3 such that i / g[B]2. It is simple to see that i = 0, so f is < -to-one onB.

The following is proved by adapting an argument of Baumgartner and Taylor [BauT].

LEMMA 7.2. Suppose J is an ideal on such that J+[J+]23, and (J,A,F) is 0-good, where

A J+ and F : 2. Then either there exists C J+ P(A) such that F is constantly 0 on

[C]2, or for every < , there exists Q A such that o.t.(Q) = and F is constantly 1 on [Q]2.

Proof. Select B J+ P(A) so that { B : F(, ) = 1} J for every B. By Lemma 7.1, thereexists S J+ P(B) so that | { S : F(, ) = 1} |< for every S. Define for < by :

(i) 0 = S;

(ii) +1 = the least < with the property that > for every S such that F(, ) = 1 forsome S ;

(iii) = 0.

Let X be the set of all limit ordinals < . For X, n and j < 2, set

dj,n = S [+2n+j , +2n+j+1).

For j < 2, letDj = {dj,n : X and n }.

Select k < 2 so that Dk J+. Notice that F(, ) = 0 if (, ) [Dk]2 and {, } dk,n for all X and n .

Define h : [Dk]23 by stipulating that h(, ) = 0 if and only if {, } dk,n for all X and

n , and h(, ) = 1 if and only if F(, ) = 1. There are W J+ P(Dk) and i < 3 so that

i / h[W]2. Clearly, i = 0. If i = 1, F is identically 0 on [W]2. Now assume i = 2. Let Z be the setof all (, n) X such that W dk,n = . Suppose that there is < such that o.t.(W d

k,n)

for every (, n) Z. Then there exists C J+ P(W) such that | C d,n |= 1 for any (, n) Z.Clearly, F takes the constant value 0 on [T]2.

PROPOSITION 7.3. Suppose (2, ) is a cardinal such that (, )2. Then

nonnonnonnonnonnonnon(J+[J+]23) nonnonnonnonnonnonnon(J

+(J+, + 1)2).

Proof. By Theorem 2.2 and Lemmas 6.3, 7.1 and 7.2.

Let us now consider the partition relation J+[J+]2. We begin with the following observation.

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PROPOSITION 7.4. Suppose is inaccessible. Then there is an ideal J on such that (a)J+/ [J+]2, (b) J is not a weak semi-Q-point, (c) aJ > , and (d) J

+(J+, )2 for every < .

Proof. Let < : < > be the increasing enumeration of all strong limit infinite cardinals < . letZ be the set of all regular infinite cardinals < . For Z, set = ()++. Then / []2 by

a result of Todorcevic [To1]. On the other hand, by a result of Erdos and Rado (see [EHMaR], Corollary17.5), (, )2 for every infinite cardinal < . Pick pairwise disjoint A for Z so that

| A |= for any Z, andZ

A = . Let J be the set of all B such that

| { Z :| B A |= } |< .

It is simple to see that J is an ideal on .

For Z, pick g : [A]2 so that g[B]2 = for every B A with | B |= . Let G : []2

be such thatZ

g G. Then clearly G[C]2 = for any C J+.

Define f by stipulating that f1({}) = A for every Z. Clearly, there is no S J+ so that

| S f1

({}|| | for all < . Hence J is not a weak semi-Q-point.

Let us next show that aJ > . Thus suppose that B J+ for < , and B B J whenever < < . Select a strictly increasing function k : Z so that

| (B (


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Remark. It is shown in [S] that if (a) is a successor cardinal 2 with 2


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that { < : g() fB()} = for every B X. Set C =

{a A : g() a}. Then C H+, and

moreover C A I, for any < .

Definition. d

, is the least cardinality of any X (P()) with the property that for every

g (P()), there is x P(X) such that g() fx f() for every < .

Remark. It is shown in [MRoS] that d

, = d cov(, +, +, ), where cov(, +, +, ) denotes the

least cardinality of any X P+() such that for every b P+(), there is x P(X) with b x.

Remark. It is immediate that I, is a weak -point. On the other hand aI, > does not necessarily

hold. In fact if cf() = and d

, for every cardinal [, ), then aI, = ([M6]).

LEMMA 8.5. Suppose that H is an ideal on P() with aH = . Then there is an ideal K on P()such that (a) K is not a weak -point, (b) cof(K) cof(H), and (c) cof(K) cof(H).

Proof. Select A H+ for < so that () A A whenever < < , and () for any C H+,there is < with C A H+. Let K be the set of all B P() such that B A H for some < . It is simple to check that K is as desired.

THEOREM 8.6.

(i) There is an ideal H on P() such that (a) aH = , (b) cof(H) = d,, and (c) cof(H) d

,.

(ii) There is an ideal K on P() such that (a) K is not a weak -point, (b) cof(K) = d,, and

(c) cof(K) d

,.

Proof. (i) : Set H = N S,. Then aH = by Corollary 8.2. Moreover, cof(H) = d, ([MPeS1]) and

cof(H) = d

, ([MRoS]).

(ii) : By (i), Lemma 8.5 and Theorem 8.4.

Remark. Theorem 8.6 is not optimal, even under GCH. In fact, suppose that the GCH holds, = +,where is a cardinal of cofinality < , and is not the successor of a cardinal of cofinality cf(). Then

d

, = ([MRoS]). Moreover, there is A (N S,)

+ such that cof(N S, | A) = ([MPeS2]). Hence there

is by Corollary 8.2 an ideal H on P() (namely H = N S, | A) such that cof(H) < d

, and aH = ,

and by Lemma 8.5 an ideal K on P() such that cof(K) < d

, and K is not a weak -point.

9. Weak -pointness

Definition. An ideal H on P() is a weak -point if given A H+ and g (P()), there isB H+ P(A) such that g((a ) b for all a, b B with (a ) < (b ).

Our primary concern in this section is with the problem of determining when I, is a weak -point.We will first give a sufficient condition and then prove that this condition is necessary if is inaccessible.

The following is proved as Lemma 2.1 in [M2].

THEOREM 9.1. Let H be an ideal on P() such that cof(H) < cov(MMMMMMM,). Then H is a weak -point.

QUESTION. Is it consistent that 2 and I,+ is a weak -point ?

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THEOREM 9.2. Suppose that for all A I+, with A {a : (a ) a}, there is B I+, P(A)

such that (a ) b for all a, b B with (a ) < (b ). Then < nonnonnonnonnonnonnon(weakly selective) forevery K(, ).

Proof. Suppose that T P( ) is such that | T P(a) |< for every a P(), and J is an idealon with cof(J) | T| . Select Dd J for d T so that for every W J, there is u P(T) {}

with W du

Dd. Now fix G J for < . Define A P() by stipulating that a A if and only

if there is < such that (a) = max(a ), (b) /

dTP(a)

Dd, and (c) / G for every a .

Let us show that A I+,. Given c P(), pick < so that /

dTP(c)

Dd and for every

c , > and / G. Set e = c {}. Then e A.

By our assumption there is B I+,P(A) such that (a) b for all a, b B with (a) < (b).

Set C = {(a ) : a B}. Then C J+. Moreover, / G for all , C with < .

We mention the following partial converse to Theorem 9.2.

PROPOSITION 9.3. Suppose that 2 (b )). Hence b / Sa,so f((a )) b .

Definition. For A P(), let

[A]2 = {((a ), b) : a, b A and (a ) < (b )}

Remark.[P()]

2 = {(, b) P() : < (b )}.

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Definition. For a, b P(), let a b just in case a b and (a ) < (b ).

Definition. For A P(), let

[A]2 = {((a ), b) : a, b A and a b}.

Remark. [P()]2 = [P()]2.

THEOREM 9.4. Suppose that is inaccessible and H is an ideal on P() such thatcof(H) < cov(MMMMMMM,), and let A H+. Then there is C H+ P(A) such that [C]2 = [C]

2.

Proof. For < , set A = {a A : (a ) = }. By induction on < , we define ck {} Afor k 2 as follows. Given k 2, set

ek =

{ck : k1({1})}

andZk = {a A : ek a}.

If Zk = , let ck be an arbitrary member of Zk. Otherwise let ck = .

Set = cof(H) and pick B H for < so that H =


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10. H+

(H+, )2

Definition. Let H be an ideal on P() and an ordinal. H+ (H+, )2 means that given

F : [P()]22 and A H+, there is B A such that either B H+ and F is identically 0 on

[B]2

or (B, ) has order type and F is identically 1 on [B]2

.

In this section we show that H+

(H+, + 1)2 for every ideal H on P() with cof(H) < cov(MMMMMMM,).

Definition. Suppose that H is an ideal on P(), A H+ and F : P()2. Then (H,A,F) is

0-good if there is D H+ P(A) such that {b D : F((a ), b) = 1} H for any a D.

The following is straightforward.

LEMMA 10.1. Suppose that (H,A,F) is 0-good, where H is an ideal on P() which is both a weak-point and a weak -point, A H+ and F : P()2. Then F is identically 0 on [C]2 for some

C H+ P(A).

Definition. Given an ideal H on P() and B H+, let MdH,B be the set of all Q H+ P(B) such

that (i) any two distinct members of Q are disjoint, and (ii) for every A H+ P(B), there is C Qwith A C H+.

LEMMA 10.2. Suppose that (H,A,F) is not 0-good, where H is an ideal on P(), A H+ and

F : P()2, and let B H+

P(A). Then there exist QB MdH,B and B : QBB such that

(i) B(D) b and F((B(D) ), b) = 1 whenever b D QB, and (ii) (B(D)) = (B(D) )

for any two distinct members D and D of QB.

Proof. Set T = {(a ) : a B} and define : T (H+ P(B)) P(B) by (, C) = {b C :F(, b) = 1}. Now using induction, define and T and B H+ P(B) for < so that :

(0) If < , B (


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We claim that {B : < } MdH,B. Suppose otherwise. Then there exists E H+ P(B) such that

E B H for every < . Since

E (


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Remark. Let H be an ideal on P(). Then

{(a ) : a A} (JH|A)+

for every A H+.


LEMMA 11.2. Given an ideal H on P(), the following are equivalent :

(i) JH is a local Q-point.

(ii) For every g , there is B H+ such that g((a )) < (b ) for all a, b B with(a ) < (b ).

Suppose is a limit cardinal. If + < non(weak Q-point), then by Lemma 11.1 JI,+

|A is a local

Q-point for every A I+,+

. The following shows that this implication can be reversed.

PROPOSITION 11.3. Suppose that is a limit cardinal and JI,|A is a local Q-point for every

A I+,. Then < non(weak Q-point) for every K(, ).

Proof. Suppose that J is an ideal on and T P() is such that cof(J) | T| and | TP(a) |< for every a P(). Select Bd J for d T so that for every D J, there is u P(T) {} with

D du

Bd. Let A be the set of all a P() such that (a ) / Bd for every d T P(a ).

It is simple to see that A I+,. Now fix g . By Lemma 11.2, there is C (I, | A)+ such that

g((a )) < (b ) for all a, b C with (a ) < (b ). Set

D = {(a ) : a C A}.

Then D J+. Moreover g() < for all , D with < . Hence J is a local Q-point.

THEOREM 11.4. Suppose that is an infinite cardinal < such that (, )2, and H is an

ideal on P() with cof(H) < non(weakly selective). Then H+

(H+, + 1)2.

Proof. Fix G : 2 and A H+. Define F : P()2 by F(, b) = G(, (b )).

First suppose (H,A,F) is 0-good. Pick D H+ P(A) so that

{b D : F((a ), b) = 1} H

for any a D. Set B = { < : G(, ) = 1} for < . Then B(a) JH|D for every a D since

D UB(a) = {b D : G((a ), (b )) = 1} = {b D : F((a ), b) = 1}.

By Lemma 11.1 cof(JH|D) < non(weak P-point) so there is G (JH|D)+ such that | G B(a) |<

for every a D. Notice that D UG H+. Select g so that (b) / B(a) for all a, b D UGsuch that g((a )) < (b ). By Lemma 11.1

cof(JH|(DUG)) < non(weak Q-point)

and hence by Lemma 11.2 there is R (H | (D UG))+

such that g((a )) < (b ) for alla, b R with (a ) < (b ). Then R D UG H+ P(A) and moreover F is identically 0 on[R D UG]2,.

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Finally, suppose (H,A,F) is not 0-good. Since aH > by Theorems 2.2 and 8.4, there is by Lemma 10.4C A such that (C, ) has order type + 1 and F is identically 1 on [C]2. It is immediate that Gis constantly 1 on [C]2,.

Remark. Suppose is a successor cardinal. Then by Theorem 11.4 + < d implies that

I+

,+

(I+

,+ , + 1)2

for every cardinal 2 such that (, )2

. Conversely, it will be shownin the next section that I+

,+

(I+,+

, 3)2 implies that + < d.

12. H+

(H+; )2H+

(H+; )2H+

(H+; )2H+

(H+; )2H+

(H+; )2H+

(H+; )2H+

(H+; )2

Definition. Given an ideal H on P() and an ordinal , H+

(H+; )2 means that for all

F : [P()]2,2 and A H+, there is B A such that either B H+ and F is identically 0 on

[B]2

,, or {(a ) : a B} has order type and F is identically 1 on [B]2

,.

Remark. H+

(H+, )2 H+

(H+, )2 H+

(H+; )2 (, )2.

We will prove that I+,

(I+,+

; )2 if and only if + < non(J+(J+, )2).

THEOREM 12.1. Suppose that 3 and H is an ideal on P() such that

cof(H) < non(J+(J+, )2). Then H+

(H+; )2.

Proof. By Lemma 11.1, (JH|A)+((JH|A)

+, )2 for every A H+. The desired conclusion easily

follows.

THEOREM 12.2. Suppose that 3 and I+,

(I+,; )2. Then < non(J+(J+, )2)

for every K(, ).

Proof. The proof is an easy modification of that of Proposition 11.3.

Remark. Suppose that is inaccessible and 3 . Then by Theorems 12.1 and 12.2,I+,

(I+,; )2 if and only if


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13. H+

(H+)2H+

(H+)2H+

(H+)2H+

(H+)2H+

(H+)2H+

(H+)2H+

(H+)2

Definition. Given an ideal H on P(), H+ (H+)2 (respectively, H+

(H+)2) means that for all

F : [P()]22 (respectively, F : [P()]2,2) and A H

+, there is B H+ P(A) such that F

is constant on [B]2 (respectively, [B]2,).

THEOREM 13.1. Suppose is weakly compact. Then H+

(H+)2 for every ideal H on P() such

that cof(H) < cov(MMMMMMM,).

Proof. Suppose that H is an ideal on P() with cof(H) < cov(MMMMMMM,), F : P()2 and A H+.

Then cof(H) < d, by Proposition 4.1 and therefore by a result of [M5] there are B H+ P(A) and

i < 2 such that{b B : F((a ), b) = i} I,

for every a B. Since H is a weak -point by Theorem 9.1, there is C H+ P(B) such that Ftakes the constant value i on [C]2.

Remark. It follows from Theorem 6.5 (ii) and Theorem 15.1 (below) that if is weakly compact, then

H+

(H+)2 for every ideal H on P() such that cof(H) < non(weakly selective).

COROLLARY 13.2. The following are equivalent :

(i) is weakly compact and


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(i) f(0) = 0 ;(ii) f( + 1) = f( f()) + 1 ;(iii) f() =

0.

Remark. f is a strictly increasing function.Remark. If g is a strictly increasing function such that g() f() for all < , then

g( f()) [ f(), f( + 1)) for every < .

Definition. Given f

( ) and a cardinal (0, ), we define cf, : f() by stipulatingthat cf, takes the constant value on [ f(), f( + 1)).

Definition. Suppose that T P( ) is such that (a) | T | d, and (b) | T P(a) |< for everya P().Let T : T

be such that given g , there is u P(T) {} such that

g() du

(T(d))()

for all < .For e P( ), let T,e =| T P(e) | and select a bijection kT,e : T,eT P(e).

Also, define fT,e by

fT,e() = max(,

dTP(e)

(T(d))()).

Finally, let AT be the set of all a P() such that (i) TP(a) = , and (ii) (a) fT,a(T,a).Remark. AT I

+,.

THEOREM 14.2. Suppose that K(, ) and d. Then I+,

/ [I+,]

2.

Proof. Select T P( ) so that | T |= and | T P(a) |< for every a P(). We define apartial function F from AT to T by stipulating that

F(, a) = kT,a(cfT,a,T,a())

if a AT and < fT,a(T,a).Now fix B I+, P(AT) and x T. Let g

be the increasing enumeration of the elements of the

set {(b ) : b B}. Select u P(T) {} so that g() du

(T(d))() for all < . Now pick

a B so that x (u) a. Notice that g() fT,a() for every . Let T,a be suchthat kT,a() = x. Then

fT,a() g( fT,a()) < fT,a( + 1) fT,a(T,a) (a ).Moreover,

F(g( fT,a()), a) = kT,a() = x.since

cfT,a,T,a(g(fT,a())) =

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15. H+

[H+]2H+

[H+]2H+

[H+]2H+

[H+]2H+

[H+]2H+

[H+]2H+

[H+]2

Definition. Given a cardinal [2, ] and an ideal H on P(), H+

[H+]2 means that for all

F : [P()]2, and A H+, there is B H+ P(A) such that F[B]2, = .

Remark. / []2 H+ / [H+]2 H

+ / [H+]2.

The following result shows that I+,+

[I+,+

]2 if and only if + < non(J

+[J+]2).

THEOREM 15.1. Let be a cardinal with 2 . Then :

(i) H+

[H+]2 for every ideal H on P() such that cof(H) < non(J+[J+]2).

(ii) If I+,

[I+,]2, then < non(J

+[J+]2) for every K(, ).

Proof. (i) : Use Lemma 11.1.(ii) : Argue as for Proposition 11.3.

Remark. Thus assuming is inaccessible, I+,

[I+,]2 if and only if


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Universite de Caen -CNRS The Hebrew UniversityMathematiques Institute of MathematicsBP 5186 91904 Jerusalem14032 CAEN CEDEX IsraelFrancee-mail : [email protected] Rutgers University

Departement of MathematicsNew Brunswick, NJ 08854USA

e-mail : [email protected]

Documents

Pierre Matet and Saharon Shelah- Positive Partition Relations for P-kappa(lambda)