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

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.


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