We solve four out of the six open problems concerning critical cardinalities of topological diagonalization properties involving τ -covers, show that the remaining two cardinals are equal, and give a consistency result concerning this remaining cardinal. Consequently, 21 open problems concerning potential implications between these properties are settled. We also give structural results based on the combinatorial techniques.1991 Mathematics Subject Classification. 03E05, 54D20, 54D80.

Abstract. We prove some restrictions on the possible cofinalities of ultrapowers of the natural numbers with respect to ultrafilters on the natural numbers. The restrictions involve two cardinal characteristics of the continuum, the splitting number s and the groupwise density number g.

Abstract. We answer a question from Raghavan and Steprāns by showing that s = s ω,ω . Then we use this to construct a completely separable maximal almost disjoint family under s ≤ a, partially answering a question of Shelah.

There are inequalities between cardinal characteristics of the continuum that are true in any model of ZFC, but without a Borel morphism proving the inequality. We answer some questions from Blass [1].

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