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Application of base tree theorem. (English) Zbl 0978.03036

The authors study ideals \({\mathcal K}^\kappa\), where \(2\leq \kappa\leq {\mathfrak c}\) is a cardinal, on \([\omega]^\omega\), which are defined as follows: For a maximal almost disjoint (MAD) family \(\mathcal A\), let \(J^\kappa({\mathcal A})\) be the set of all \(X\in[\omega]^\omega\) which meet at least \(\kappa\) elements of \(\mathcal A\) infinitely often. Then \({\mathcal K}^\kappa\) is the ideal generated by all sets \(J^\kappa({\mathcal A})\), where \(\mathcal A\) is a MAD family. Among other things, the additivity, covering and cofinality numbers of these ideals are studied. Also, the solution to a question of Hechler is given.

MSC:

03E05 Other combinatorial set theory
03E17 Cardinal characteristics of the continuum
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