Strong partition properties for infinite cardinals

Abstract

The notion of a “partition relation”, as it has been studied in the context of set theory for the past several years, was inspired by the following theorem of F. P. Ramsey [14]: Theorem 0.1. Let n be a positive integer and let {A, B} be a partition of those subsets of the nonnegative integers containing exactly n elements. Then there exists an infinite subset x of the nonnegative integers all of whose n-element subsets are contained in only one of A or B. (Any such set x is said to be “homogeneous” for the partition.)