Cardinal invariants of the continuum and combinatorics on uncountable cardinals

Abstract

We explore the connection between combinatorial principles on uncountable cardinals, like stick and club, on the one hand, and the combinatorics of sets of reals and, in particular, cardinal invariants of the continuum, on the other hand. For example, we prove that additivity of measure implies that Martin’s axiom holds for any Cohen algebra. We construct a model in which club holds, yet the covering number of the null ideal cov ( N ) is large. We show that for uncountable cardinals κ ≤ λ and F ⊆ [ λ ] κ , if all subsets of λ either contain, or are disjoint from, a member of F , then F has size at least cov ( N ) etc. As an application, we solve the Gross space problem under c = ℵ 2 by showing that there is such a space over any countable field. In two appendices, we solve problems of Fuchino, Shelah and Soukup, and of Kraszewski, respectively.