shimon garti and saharon shelah- strong polarized relations for the continuum

8/3/2019 Shimon Garti and Saharon Shelah- Strong Polarized Relations for the Continuum

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STRONG POLARIZED RELATIONS FOR THE

CONTINUUM

SHIMON GARTI AND SAHARON SHELAH

Abstract. We prove that the strong polarized relation`2

`2

1,12

is consistent with ZFC. We show this for = 0, and for every super-compact cardinal . We also characterize the polarized relation b elowthe splitting number.

2000 Mathematics Subject Classification. 03E05, 03E55, 03E35.Key words and phrases. Partition calculus, forcing, large cardinals.First typed: February 2010

Research supported by the United States-Israel Binational Science Foundation. Publica-tion 964 of the second author.

1


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2 SHIMON GARTI AND SAHARON SHELAH

0. introduction

The balanced polarized relation

1,1

2asserts that for every color-

ing c : 2 there are A and B such that otp(A) = , otp(B) = and c (A B) is constant. This relation was first introduced in [2], andinvestigated further in [1]. A wonderful summary of the basic facts for thisrelation, appears in [7].Apparently, this relation can be true only when and . It means

that the strongest form of it is the case of

1,12

. We can give a name

to this situation:

Definition 0.1. The strong polarized relation.

If

1,1

then we say that the pair (, ) satisfies the strong polarized

relation with colors.

If 2 = + then+

+

1,12

. This result, and similar negative results,go back to Sierpinsky. Despite the negative results under the (local) assump-

tion of the GCH, we can show that the positive relation+

+

1,12

isconsistent with ZFC. Such a result appears in [3], and it was known forthe specific case of = 0 (under the appropriate assumption, e.g., MA +20 > 1, see Laver in [4] which proves that if Martins Axiom holds for

then

1,12

).So the negative result under the GCH cannot become a theorem in ZFC.

But we can restate this result in the form2

2

1,12

. In this light, a

natural question is whether the positive relation2

2

1,1

2

is consistentwith ZFC. One might suspect that the answer is negative, and this is thecorrect generalization of the negative result under the GCH. Notice that theresult of Laver (from [4]) does not help in this case, since MA20 never holds.

Moreover, in the first section we prove that if 0 < cf() < s, then

1,12

. Hence, for every below the continuum (with uncountable

cofinality), we can force a positive result by increasing s. But s 20 , so thismethod does not help in our question. Again, under some assumptions onecan prove negative results. For example, if 2 is regular and MA(countable)

holds, then

2

2

1,12

.Nevertheless, we shall prove that a positive relation is consistent here.

In the first section we deal with = 0

. Here we can use a finite supportiteration of ccc forcing notions, yielding a ccc forcing notion in the limitof the sequence. We indicate that cf (2) 2 in our construction, and

it might be that the relation2

2

1,12

is provable in ZFC whenevercf(2) = 1. Notice that cf(2

) = 1 implies the existence of weak diamondon 1, so the negative relation becomes plausible.

In the second section we try to generalize it to higher cardinals. Here weencounter some difficulty, since the chain condition is not inherited from themembers of the iteration. Starting with Laver-indestructible supercompact


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STRONG POLARIZED RELATIONS FOR THE CONTINUUM 3

cardinal, we can overcome this problem (as well as other obstacles in thegeneralization of the countable case).

We try to use standard notation. We use the letters ,,,, for infinitecardinals, and ,,,,, for ordinals. For a regular cardinal we denotethe ideal of bounded subsets of by Jbd . For A, B we say that A

B

when A \ B is bounded in . The symbol [] means the collection of all thesubsets of of cardinality . We denote the continuum by c.

Recall that by Laver (in [5]) we can make a supercompact cardinal indestructible, upon forcing with -closed forcing notions. We shall use thisassumption in 2. We indicate that p q means (in this paper) that q givesmore information than p in forcing notions.

An important specific forcing to be mentioned here is the Mathias forcing.Let D be a nonprincipal ultrafilter on . We define MD as follows. Theconditions in M

Dare pairs of the form (s, A) when s []


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1. The countable case

We prove, in this section, the consistency of c

c

1,1

n, for every

natural number n. The general pattern of the proof will be used also in thenext section.

Theorem 1.1. A positive realtion for c.

The strong relationc

c

1,1n

is consistent with ZFC, for every natural

number n.

Proof.Choose an uncountable cardinal so that 0 = and cf() 2. We definea finite support iteration Pi,Q

j : i 1, j < of ccc forcing notions, such

that |Pi| = for every i 1.

Let Q0 be (a name of) a forcing notion which adds reals (e.g., Cohenforcing). For every j < 1 let D

j be a name of a nonprincipal ultrafilter on

. Let Q

1+j be a ccc forcing notion which adds an infinite set Aj such

that (B Dj)(A

j B A

j \ B). The Mathias forcing MD

j

canserve. At the end, set P =

{Pi : i < 1}.

Since every component satisfies the ccc, and we use finite support itera-tion, P is also a ccc forcing notion and hence no cardinal is collapsed in VP.In addition, notice that 20 = after forcing with P. Our goal is to prove

that

1,1n

in VP.

Let c

be a name of a function from into n. For every < 20 wehave a name (in V) to the restriction c

({} ). P is ccc, hence the color

of every pair of the form (, n) is determined by an antichain which includesat most 0 conditions. Since we have to decide the color of 0-many pairsin c

({} ), and the length ofP is 1, we know that c

({} ) is a

name in Pi() for some i() < 1.For every j < 1 let Uj be the set { < : i() j}. Recall that

1 < 2 cf(), hence for some j < 1 we have Uj []. Choose such

j, and denote Uj by U. We shall try to show that U can serve (after someshrinking) as the first coordinate in the monochromatic subset.

Choose a generic subset G P, and denote Aj [G] by A. For each U

we know that c ({} A) is constant, except a possible mistake over a

finite subset of A. But this mistake can be amended.For every U choose k() and m() < n so that ( A)[

k() c

[G](, ) = m()]. n is finite and cf() > 0, so one can fixsome k and a color m < n such that for some U1 [U]

we have U1 k() = k m() = m.

Let B be A \ k, so B [] . By the fact that U1 U we know thatc(, ) = m for every U1 and B, yielding the positive relationc

c

1,1n

, as required.1.1


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Remark 1.2. Assume is an uncountable regular cardinal. Denote by

st

1,1

n

the assertion that for every coloring c : n there

exists B [] and a stationary subset U so that c U B is constant.

Our proof gives the consistency ofc

st

c

1,1n

, when the continuum isregular.

Let us turn to cardinals below the continuum. We deal with the relation

1,12

, when < s. We shall prove that this relation holds iff is ofuncountable cofinality.

Recall:

Definition 1.3. The splitting number.Let F = {S : < } be a family of subsets of , and B []

.

() F splits B if |B S| = |B \ S| = 0 for some <

() F is a splitting family if F splits B for every B []() s = min{|F| : F is a splitting family}

Claim 1.4. The polarized relation below s.

Assume < s.

Then

1,12

iff cf() > 0.

Proof.Suppose first that cf() > 0. Let c : 2 be a coloring. DefineS = {n : c(, n) = 0} for every < , and F = Fc = {S : < }.Since < s, we know that F is not a splitting family.

Choose an evidence, i.e., B [] which is not splitted by F. It means

that (B

S) (B

\ S) for every < . At least one of this twoopstions occurs -many times, and since all we need for the first coordinate(in the monochromatic subset) is its cardinality, we shall assume (withoutloss of generality) that B S for every < .

For every < there exists a finite set t such that B \ t S.There are countably-many t-s, and cf() > 0, so for some t []


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We indicate that for s itself we believe that the relations

s

1,12

isindependent of ZFC. We hope to shed light on this issue in a subsequent

work.


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2. The supercompact case

In this section we prove the consistency of2

2

1,1

2for every super-

compact . We shall use [5] for making indestructible (in fact, all we needis the measurability of at every stage of the iteration), and [6] for preserv-ing the property of -cc along the iteration. We shall use a generalizationof the Mathias forcing, so we need the following:

Definition 2.1. The generalized Mathias forcing.Let be a supercompact (or even just measurable) cardinal, and D a non-principal -complete ultrafilter on . The forcing notion MD consists ofpairs (a, A) such that a []


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give different values, since P is +-cc. But P is of length +, so c appears

at some early stage P().

For every < +, set U = { < : () }. Since cf() = ++ > +,one can pick an ordinal < + so that |U| = . Let U be U, and let G Pbe generic over V. Denote A

[G] by A.

For every U, c is constant on A, except a small (i.e., of cardinality

less than ) subset of A. For each U choose () and () suchthat () A c

[G](, ) = (). By the assumptions on , and

the cofinality of , there is U1 [U] and , such that ()

and () for every U1.Set B = A \ , and notice that |B| = . By the above considerations,

for every U1 and every B we have c(, ) = . Hence

1,1

,

and the proof is complete.

2.2

Remark 2.3. Notice that the relation

1,12

is completely determined

under GCH, as follows from [7] theorem 4.2.8.


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References

1. P. Erdos, A. Hajnal, and R. Rado, Partition relations for cardinal numbers, Acta Math.

Acad. Sci. Hungar. 16 (1965), 93196. MR MR0202613 (34 #2475)2. P. Erdos and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62

(1956), 427489. MR MR0081864 (18,458a)3. Shimon Garti and Saharon Shelah, A strong polarized relation, in preperation.4. R. Laver, Partition relations for uncountable cardinals 20 , Infinite and finite sets

(Colloq., Keszthely, 1973; dedicated to P. Erdos on his 60th birthday), Vol. II, North-Holland, Amsterdam, 1975, pp. 10291042. Colloq. Math. Soc. Jan os Bolyai, Vol. 10.MR MR0371652 (51 #7870)

5. Richard Laver, Making the supercompactness of indestructible under-directed closedforcing, Israel J. Math. 29 (1978), no. 4, 385388. MR MR0472529 (57 #12226)

6. S. Shelah, A weak generalization of MA to higher cardinals, Israel J. Math. 30 (1978),no. 4, 297306. MR MR0505492 (58 #21606)

7. Neil H. Williams, Combinatorial set theory, studies in logic and the foundations of

mathematics, vol. 91, North-Holland publishing company, Amsterdam, New York, Ox-ford, 1977.

Institute of Mathematics The Hebrew University of Jerusalem Jerusalem

91904, Israel

E-mail address: [email protected]

Institute of Mathematics The Hebrew University of Jerusalem Jerusalem

91904, Israel and Department of Mathematics Rutgers University New Brunswick,

NJ 08854, USA

E-mail address: [email protected]

URL: http://www.math.rutgers.edu/~shelah

shimon garti and saharon shelah- strong polarized relations for the continuum

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