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How to drive our families mad

Sakaé Fuchino·Stefan Geschke·Osvaldo Guzman·Lajos Soukup
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Abstract

Given a family $F$ of pairwise almost disjoint sets on a countable set $S$, we study maximal almost disjoint (mad) families $F^+$ extending $F$. We define $a^+(F)$ to be the minimal possible cardinality of $F^+\setminus F$ for such $F^+$, and $a^+(κ)=\sup\{a^+(F): |F| \leq κ\}$. We show that all infinite cardinal less than or equal to the continuum continuum can be represented as $a^+(F)$ for some almost disjoint $F$ and that the inequalities $\aleph_1=a<a^+(\aleph_1)=c$ and $a=a^+(\aleph_1)<c$ are both consistent. We also give a several constructions of mad families with some additional properties.

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Sakaé Fuchino, Stefan Geschke, Osvaldo Guzman, Lajos Soukup. 2017-03-07. How to drive our families mad. https://arxiv.org/abs/math/0611744

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