shimon garti and saharon shelah- a strong polarized relation

8/3/2019 Shimon Garti and Saharon Shelah- A Strong Polarized Relation

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A STRONG POLARIZED RELATION

SHIMON GARTI AND SAHARON SHELAH

Abstract. We prove that the strong polarized relation`+

`+

1,12

is consistent with ZFC, for a singular which is a limit of measurablecardinals (but in fact, it follows from pcf assumptions).

2000 Mathematics Subject Classification. 03E05, 03E55.Key words and phrases. Partition calculus, cardinal arithmetic, large cardinals.First typed: December 2008

Research supported by the United States-Israel Binational Science Foundation. Publica-tion 949 of the second author.

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2 SHIMON GARTI AND SAHARON SHELAH

0. introduction

The polarized relation

0 1

0 1

1,1asserts that for every coloring

c : 2 there are A and B such that either otp(A) =0, otp(B) = 0 and c (A B) = {0} or otp(A) = 1, otp(B) = 1 andc (A B) = {1}. This relation was first introduced in [3], and investigatedfurther in [2].If (0, 0) = (1, 1) then we get the so-called unbalanced form of the relation.The balanced form is the case (0, 0) = (1, 1), and in this case we can write

also

1,12

(stipulating = 0 = 1 and = 0 = 1). With this

shorthand, the notation

1,1

means the same thing, but the number

of colors is instead of 2.From some trivialities and simple limitations, it follows that the case = +

and = is interesting, for an infinite cardinal . It is reasonable todistinguish between three cases - is a successor cardinal, is a limitregular cardinal (so it is a large cardinal) and is a singular cardinal; weconcentrate in the latter case.By a result of Cudnovskii in [1], if is measurable then the relation

+

+

1,1holds in ZFC for every < + (see also [4], for discussion on

weakly compact cardinals). In a sense, this is the best possible result, since

we know that the assertion

+

+

1,12

is valid under the GCH for

every infinite cardinal (see [13]). This limitation gives rise to the following

problem: Can one prove that the strong relation

+

+

1,12

is consistent

with ZFC?We give a positive answer. For a singular which is a limit of measurables,

we can show that under some cardinal arithmetic assumptions (including

the violation of the GCH, of course) one can get

+

+

1,1

for every

< cf(). This result is stronger, on one hand, than the balanced result+

1,1

which is proved in [12] for every < +. On the other hand,

the result there is proved in ZFC, whence the strong relations in this papercan not be proved in ZFC.One can view this result as the parallel to the ordinary partition relationwith respect to weakly compact cardinals. We know that if is inaccessiblethen (, )2 for every < , but the strong (and balanced) relation

()22 kicks up in the chart of large cardinals, making it weakly compact.The result here is similar, replacing the ordinary partition relation by thepolarized one.

Our notation is standard. We use the letters ,,, for infinite cardinals,and ,,,, ,,i ,j for ordinals. For a regular cardinal we denote theideal of bounded subsets of by Jbd . For A, B we say A

B whenA \ B is bounded in ; the common usage of this symbol is for = 0, buthere we apply it to uncountable cardinals.


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A STRONG POLARIZED RELATION 3

The product


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1. The combinatorial theorem

We state the main result of the paper:

Theorem 1.1. The main result.Let be a singular cardinal, = cf() and < .Assume 2 < cf() < 2.Suppose is a limit of measurable cardinals, = : < is a sequenceof measurables with limit so that < 0,


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2. forcing preliminaries

We need some preliminaries, before proving the main claim of the nextsection. First of all, we shall use a variant of Lavers indestructibility (see[6]), making sure that a supercompact cardinal will remain supercompactupon forcing with some prescribed properties. Let us start with the followingdefinition:

Definition 2.1. Strategically completeness.Let P be a forcing notion, and p P.

(a) The game (p,P) is played between two players, com and inc. Itlasts moves. In the -th move, com tries to choose p P suchthat p P p and < q P p. After that, inc tries to chooseq P such that p P q.

(b) com wins a play if he has a legal move for every < .(c) P is -strategically complete if the player com has a winning strat-egy in the game (p,P) for every p P.

Claim 2.2. Indestructible supercompact and strategically completeness.Let be a supercompact cardinal in the ground model.There is a forcing notionQ which makes indestructible under every forcingP with the following properties:

(a) P is -strategically complete for every < (b) , andP H()(c) for some : V M such that = crit(), M M and for every

G P which is generic over V, we have M[G] |= {(p) : p G}

has an upper bound in (P).

Proof.Basically, the proof walks along the line of [6], using Lavers diamond. Inthe crux of the matter, when Laver needs the -completeness, we employrequirement (c) above.

2.2We define now the single step forcing notion Q to be used in the proof of3.3. This is called the -dominating forcing (it appears also in [9]). We willuse an iteration which consists, essentially, of these forcing notions:

Definition 2.3. The -dominating forcing.Let be a supercompact cardinal.Suppose = : < is an increasing sequence of cardinals so that2||+0 < < for every < .

() p Q iff:(a) p = (, f) = (p, fp)(b) g() < (c)

{ : < g()}

(d) f

{ : < }(e) f


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() p Q q iff (p,q Q and)(a) p q

(b) fp() fq(), for every <

Notice that if g(p) < g(q ) then fp() q(), since fp() fq() = q (). The purpose of Q is to add (via the generic object) adominating function in the product of the -s.

Observation 2.4. Basic properties ofQ.LetQ be the -dominating forcing (for the supercompact cardinal ).

(a) Q satisfies the +-cc.

(b) Q is -strategically complete for every < .

Proof.

(a) If p = (, fp), q = (, fq), define f() = max{fp(), fq ()} for every < , and then r = (, f). Clearly, r Q and p, q r. So thecardinality of an antichain does not exceed the number of possible-s, which is since g() < and


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Definition 2.6. The iteration.Let be a supercompact cardinal.

Let P be the (< )-support iteration P,Q : , < , where each

Q

is (a P-name of) the forcing Q.

We would like to show that the nice properties of each component ensuredby 2.4 are preserved in the iteration. Now, the strategically completeness ispreserved, but the chain condition may fail. Nevertheless, in the case of thedominating forcing Q it holds:

Definition 2.7. Linked forcing notions.Let P be a forcing notion.P is -2-linked when for every subset of conditions {p : <

+} P thereare C, h such that:

(1) C is a closed unbounded subset of +

(2) h : + + is a regressive function(3) for every , C, if cf() = cf() = and h() = h() then

p p; moreover, p p is a least upper bound

Remark 2.8. IfQ is -2-linked, then Q is +-cc.

Lemma 2.9. Preservation of the -2-linked property.Assume =


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For proving that cf(


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3. Cardinal arithmetic assumptions

We phrase two theorems, which we shall prove in this section. The firstone asserts that there exits (i.e., by forcing) a singular cardinal, limit ofmeasurable cardinals, with some properties imposed on the product of thesemeasurables and their successors. Related works, in this light, are [5] and[10]. The second theorem deals with properties of the product of normal(uniform) ultrafilters on these cardinals.We start with the following known fact:

Lemma 3.1. Cofinality preservation under Prikry forcing.Let U be a normal (uniform) ultrafilter on a measurable cardinal .LetQU be the Prikry forcing (with respect to and U) and n : n thePrikry sequence.

Suppose = cf() = , F : Reg and cf(


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Ap, as an intersection of at most = |[]


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Levy(+, 2) upon noticing that the local GCH on reflects down to enoughmeasurables below, and the iteration P does not affect the measurability of

the cardinals below , since it does not add new bounded subsets).Let U be a normal (uniform) ultrafilter on in VP. Let QU be the Prikry

forcing applied to , adding the cofinal Prikry sequence n : n < . InVPQU we know that cf() = 0. We indicate that using Magidors forcing(from [7]), we can get a similar result for cf() = > 0.

Now, if is an increasing sequence as in definition 2.3, we know that

cf(


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A


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References

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2. P. Erdos, A. Hajnal, and R. Rado, Partition relations for cardinal numbers, ActaMath. Acad. Sci. Hungar. 16 (1965), 93196. MR MR0202613 (34 #2475)

3. P. Erdos and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62(1956), 427489. MR MR0081864 (18,458a)

4. Matthew Foreman and Andras Hajnal, A partition relation for successors of largecardinals, Math. Ann. 325 (2003), no. 3, 583623. MR MR1968607 (2004b:03063)

5. Moti Gitik and Saharon Shelah, On densities of box products, Topology Appl. 88(1998), no. 3, 219237. MR MR1632081 (2000b:03183)

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(1978), no. 1,6171. MR MR0465868 (57 #5754)8. S. Shelah, A weak generalization of MA to higher cardinals, Israel J. Math. 30 (1978),

no. 4, 297306. MR MR0505492 (58 #21606)9. Saharon Shelah, On con(dominating > cov(meagre)), preprint.

10. , More on cardinal arithmetic, Arch. Math. Logic 32 (1993), no. 6, 399428.MR MR1245523 (94j:03100)

11. , Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press Ox-ford University Press, New York, 1994, , Oxford Science Publications. MR MR1318912(96e:03001)

12. , A polarized partition relation and failure of GCH at singular strong limit,Fund. Math. 155 (1998), no. 2, 153160. MR MR1606515 (99b:03064)

13. Neil H. Williams, Combinatorial set theory, studies in logic and the foundations ofmathematics, vol. 91, North-Holland publishing company, Amsterdam, New York,

Oxford, 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]: http://www.math.rutgers.edu/~shelah

shimon garti and saharon shelah- a strong polarized relation

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