Mad families, splitting families and large continuum. (English) Zbl 1215.03061

Brendle, Jörg; Fischer, Vera

J. Symb. Log. 76, No. 1, 198-208 (2011).

The authors study the bounding, the splitting and the almost disjointness numbers \(\mathfrak{b}\), \({\mathfrak s}\) and \(\mathfrak{a}\). The bounding and the splitting numbers are independent. Baumgartner and Dordal obtained the consistency of \(\mathfrak{s} < \mathfrak{b}\) and Shelah showed the consistency of \(\mathfrak{s} < \mathfrak{b}\) using a proper forcing notion. Thus, it is of interest to obtain models in which \(2^{\aleph_0} \geq \aleph_3\). Using finite support iteration of ccc posets, Brendle obtained the consistency of \(\mathfrak{b = \kappa < a = \kappa^+}\), and Fischer and Steprāns obtained the consistency of \(\mathfrak{b = \kappa < s = \kappa^+}\).
Here, the authors show the consistency of \(\mathfrak{b = a = \kappa < s = \lambda}\), where \(\kappa < \lambda\) are arbitrary regular uncountable cardinals. Further on, they show the consistency of \(\mathfrak{b = \kappa < a = s = \lambda}\), where \(\kappa < \lambda\) are arbitrary regular cardinals above a measurable cardinal \(\mu\). The proofs are given by finite support iterations of ccc posets. The constructions use the idea of matrix iteration introduced by Blass and Shelah.

Reviewer: Martin Weese (Potsdam)

MSC:

03E17	Cardinal characteristics of the continuum
03E35	Consistency and independence results
03E55	Large cardinals

Keywords:

almost disjointness number; splitting number; bounding number

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