Some Remarks on Polarized Partition Relations

Abstract

AbstractThis paper deals with two notions: a polarized partition relations $$\left( \begin{array}{c} \alpha \\ \beta \end{array} \right) \rightarrow \left( \begin{array}{cc} \gamma &{} \eta \\ \delta &{} \lambda \end{array} \right) $$ α β → γ η δ λ 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 $$\left( \begin{array}{c} \alpha \\ \beta \end{array} \right) \rightarrow \left( \begin{array}{cc} \gamma &{} \eta \\ \delta &{} \lambda \end{array} \right) $$ α β → γ η δ λ is a consequence of the existence of product of generalized strong sequences.