Weak strong partition cardinals

Extract

In a series of papers [K2], [K3], [K4], E. M. Kleinberg established the extensive properties of what are now called “strong partition cardinals”, cardinals satisfying for all λ < κ. The purpose of this note is to show that all these consequences and the results in [H] and [W] can be obtained from the weaker relation and many from .

We assume the reader is generally familiar with Kleinberg's machinery and with the definition of . We recall that a cardinal κ satisfies iff for every partition F: [κ]^κ → A there is a p ∈ [κ]^κ such that F″ [p]^κ ≠ A. We take the liberty of regarding a p ∈ [κ]^κ both as a subset of κ and as a function from κ to κ. We assume DC throughout.

§1. From. Our results stem from the observation that the proofs in the papers cited above only require homogeneous sets for certain classes of partitions.

Definition. A partition F: [κ]^κ → λ, λ < λ, is called clopen if for all p ∈ [κ]^κ there is an α < κ such that whenever -clopen is the assertion that all clopen partitions have homogeneous sets (Spector-Watro).