Some Remarks on Polarized Partition Relations

This paper deals with two notions: a polarized partition relations αβ→γηδλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \begin{array}{c} \alpha \\ \beta \end{array} \right) \rightarrow \left( \begin{array}{cc} \gamma &{} \eta \\ \delta &{} \lambda \end{array} \right) $$\end{document} and product of generalized strong sequences. Strong sequences were introduced by Efimov in 1965 as a useful tool for proving famous theorems in dyadic spaces, i.e. continuous images of Cantor cube. In this paper we introduce the notion of product of generalized strong sequences and give the pure combinatorial proof that αβ→γηδλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \begin{array}{c} \alpha \\ \beta \end{array} \right) \rightarrow \left( \begin{array}{cc} \gamma &{} \eta \\ \delta &{} \lambda \end{array} \right) $$\end{document} is a consequence of the existence of product of generalized strong sequences.


Introduction and historical background
The notion of partition relations was introduced in [5] by Erdös and Rado as the ordinary partition relations which concerned partitions of finite subsets of a set of a given size and the polarized partition relations which concerned partitions of finite subsets of products of set of a given size, (where size means cardinality of a set or order type of an ordered set -we will specify it in the concrete situations).However, this topic has its origin in paper the Ramsey's paper [21].The main result of [21] was generalized in 1942 by Erdös, [7].
Papers that deserve attention in this topic are undoubtedly [6,9,26], however a great many new results were proved by researchsers in the recently time.This shows that the topic is extremely lively and still worth exploring.
In paper [16], we proved a several theorems which we call Ramsey type contain an alternative in the thesis: either we obtain a set of large cardinality with a certain property, or we obtain a set with small cardinality with an opposite property.This lead us to the following array notation (α) → (β, γ) n which means that for given cardinals α, β, γ and for each set A of cardinality α and a function c: [A] n → 2 there exists a set A 0 ⊆ A of cardinality β such that c|[A 0 ] 2 = {0} or there exists a set A 1 ⊆ A of cardinality γ such that c|[A 1 ] 2 = {1}.In the literature there are known significant results of such theorems.We recall here some of them.The result by Hajnal [8] says that if 2 κ = κ + , then Todorcevic, in [22], proved that PFA (Proper Forcing Axiom) implies for all α < ω 1 .Dushnik and Miller in [2] showed that for every infinite cardinal κ κ → (κ, ω) 2 .
In the literature one can meet with "a kind of combination" of the above two partition relations, i.e.
The main result of this paper is to prove that the following theorem is the consequence Theorem 1 in Section 3 which is also in [19].To make this work self-sufficient we cite the proof of theorem on product strong sequences from [19].This paper is a continuation of [19] in which we show that the polarized partition relation γ is equivalent to the existence of strong sequences.The strong sequences method was introduced by Efimov in [3] as a useful tool for proving theorems in dyadic spaces, (i.e.continuous images of the Cantor cube).Among others, Efimov showed that strong sequences does not exist in the general Cantor discontinua.The topic of strong sequences was considered by Turzański in the 90s' of the last century.Turzański reformulated the definition of strong sequences as follows.
Let X be a set and let B ⊆ P (X) be a family of non-empty subsets of X closed with respect to finite intersections.Let H α ⊆ B and S α ⊆ B such that S α is finite.A sequence (S α , H α ) α<κ is called a strong sequence iff S α ∪ H α is centered and S β ∪ H α is not centered, whenever β > α.
In [23] the author proved the following theorem on strong sequences: if there exists a strong sequence (S α , H α ) α<(κ λ ) + such that |H α | κ for all α < κ λ ) + then the family B contains a subfamily of cadinality λ + consisting of pairwise disjoint sets.
Based on this result Turzański estimated the weight of regular spaces.In [24] he gave a new proof of Esenin-Volpin Theorem of weight of dyadic spaces (in general form in class of thick spaces which possesses special subbases) and in [25] he showed that the theorem on strong sequences is equivalent to Erdös-Rado Theorem.
The investigations on strong sequences have been continued, extended and improved by the author of this paper in [13,14,15,16,17,18,19].
In paper [15] there is proved the generalization of theorem on strong sequences and it is shown that it is equivalent to the generalized Erdös-Rado Theorem.Further generalizations of these results are given in [16].In [13,18] there is introduced the cardinal invariant associated with strong sequences and there are shown some inequalities between it and well known cardinal invariants.In [17] there is shown that the existence of so called K-Lusin sets is equivalent to the existence of strong sequences of the same cardinality.The newest result concerning strong sequences are concentrated around product of generalized strong sequences and its connections with polarized partition relations, ( [19]).However, we know a number of consequences of the existence of strong sequences the topic seems not to be exhausted.The main problem followed from Efimov result is still open: if strong sequences does not exist in general Cantor discontinua for which spaces does they exist?
The paper is organized as follows.In Section 2 there are given basic definitions needed in further parts of the text.In section 3 there is proved the theorem on product of strong sequences.In Section 4 there is shown the equivalnece of Theorem 1 with polarized partition relation.
In this paper there are used standard notation and terminology.For the definitions and facts not cited her we refer the reader to [4,11].

Definitions
In the whole paper we use Greek letters to denote the cardinal or ordinal numbers, (which one we will mean at the particular parts will be follow from the context).

The polarized partition relation
means that the following statement is true: for every set A k of cardinality α k , (1 k n) and for every function 2.2.Let (X k , r k ) be sets with two-place relations r k , (1 k n).
In the whole paper we restrict our considerations to finite products of sets, because we do not need more in this moment, but the results presented in further parts of this paper can be generalized for infinite products, (with extreme caution as is usual with infinite product operations). Let We say that A ⊆ X is κ-directed if every subset of A of cardinality less than κ has a bound.
Every inaccessible cardinal is weakly inaccessible.If the GCH holds then every weakly inaccessible cardinal τ is inaccessible.Inaccessible cardinals cannot be obtained from smaller cardinals by the usual set-theoretical operations.This is one of the themes of set theory of large cardinals.The existence of inaccessible cardinals is not provable in ZFC.Moreover, it cannot be shown that the existence of inaccessible cardinals is consistent with ZFC, (see [11,Theorem 12.12]).The least inaccessible cardinal is not measurable, (see [10]).Inaccessible cardinals were introduced by Sierpiński and Tarski in 1930.

Theorem on product κ-strong sequences
The special case of the following theorems (for k = 1) was proved in [16].
for all α < η, then there exists a product κ-strong sequence Proof.Fix n < ω.Let {H α : α < η} be such that is a κ-strong sequence.Fix α and name it α 0 , (without the loss of generality one can assume that α 0 = 0).For each k, (1 k n), consider a function and let be a family of pairwise disjoint sets.Before the continuation of the proof we make the following observation.By definition of product κ-strong sequence for each α > α 0 there exists a (k, α 0 , α)-destroyer.Since we consider only k such that (1 k n) some of them must occur at least β k -times, (ω β k ≪ η k ).Now, we are ready to continue the proof.
For every relevant k we will construct inductively a) an increasing subsequence Assume that we have constructed increasing subsequence {α γ : γ < β k } of η k and families A k αγ as was done above.Next, choose α > α γ , (α < η k ) such that there exists a (k, α γ , α)- <κ k and denote this α by α δ , where δ = γ + 1.For each A k αγ ∈ A k αγ define a function be a family of pairwise disjoint sets.If δ is limit, we consider The induction step is complete.Now define a sequence Then, there would exist and β k , η k be regulars.Then either X contains a κ-directed subset of cardinality η or X contains a subset of cardinality β consisting of elements which are pairwise disjoint.
Proof.Without the loss of generality we can assume that X k ⊆ η k , (1 k n).Suppose that each κ k -directed subset of X k has cardinality less than η k .We will use Theorem 1 to prove the second alternative of the statement.
In order to do this we will construct inductively a product κ-strong sequence {H α : α < η}.
Assume that for α < η the product κ-strong sequence and put Declaration The author have no conflicts of interest to declare.
β n in the following way: Thus, we have defined at least one product κ-strong sequence {T αγ : α γ < β} of the required property.Suppose now, that at least one of product κ-strong sequence