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Critical cardinalities and additivity properties of combinatorial notions of smallness. (English) Zbl 1052.03026

\({\mathfrak p}\) denotes the minimal cardinality of a centered family \(F\subseteq[\omega]^\omega\) with no pseudointersection. \({\mathfrak t}\) denotes the minimal size of a tower on \(\omega\). One of the most important open problems in infinite combinatorics asks whether it is provable that \({\mathfrak p}= {\mathfrak t}\).
The authors deal with two types of combinatorial questions which arise from this problem. (1) They introduce two new cardinals and express them in terms of well-known cardinal characteristics of the continuum. (2) They study additivity numbers for diagonalizations of covers.
Their results give new insights into the structure of eventual dominance on the Baire space and the almost dominance ordering of the Rothberger space.

MSC:

03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results
03E05 Other combinatorial set theory
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