How to drive our families mad

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 $a^{+}(\kappa )=\sup \{a^{+}(F):|F|\le \kappa \}$. 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.

revised and extended version, 19 pages