Abstract
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).
Publisher
Cambridge University Press (CUP)
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1. Magidor-like and radin-like forcing;Annals of Pure and Applied Logic;1983-10