Some results concerning polarized partition relations.

摘要

A new way of characterizing many partition relations in terms of matrices of sets is described. Using one such characterization, we then prove a generalization of a theorem of Prikry which stated that ℵ1 ℵ2 ↛ ℵ1 ℵ0 ℵ1 is consistent with ZFC.; We then consider a new type of partition relation, which has characteristics of both polarized and non-polarized partition relations. Exploring this notion, we prove, among other things, that if each pair of distinct ordinals below ℵw is colored either blue or red, then there exist subsets H 1 and H2 of ℵw of size ℵw such that all the pairs (alpha, beta) ∈ H 1 x H2 with alpha beta are of the same color. We also prove a number of negative results concerning this notion, among them the fact that there exists a partition f : ℵ1xℵ1→ℵ 1 such that, given any subsets H1 and H2 of ℵ1 of size ℵ1 , the image of {lcub}(alpha, beta) ∈ H1 x H2 : alpha beta{rcub} under f is ℵ1 . In addition to these two results, we present a series of theorems which explore the usefulness of the new partition relations.