How to drive our families mad

Sakaé Fuchino, Stefan Geschke and Lajos Soukup

Given a family ${\mathcal F}$ of pairwise almost disjoint (ad) sets on a countable set $S$, we study families $\tilde{{\mathcal F}}$ of maximal almost disjoint (mad) sets extending ${\mathcal F}$.

We define ${\mathfrak{a}}^+({\mathcal F})$ to be the minimal possible cardinality of $\tilde{{\mathcal F}}\setminus {\mathcal F}$ for such $\tilde{{\mathcal F}}$ and ${\mathfrak{a}}^+(\kappa)=\max\{{\mathfrak{a}}^+({\mathcal F})\,:\,\mathopen{\vert\,}{\mathcal F}\mathclose{\,\vert}\leq\kappa\}$. We show that all infinite cardinal less than or equal to the continuum ${\mathfrak{c}}$ can be represented as ${\mathfrak{a}}^+({\mathcal F})$ for some ad ${\mathcal F}$ and that the inequalities $\aleph_1={\mathfrak{a}}<{\mathfrak{a}}^+(\aleph_1)={\mathfrak{c}}$ and ${\mathfrak{a}}={\mathfrak{a}}^+(\aleph_1)<{\mathfrak{c}}$ are both consistent.

We also give a several constructions of mad families with some additional properties.

Key words and phrases:almost disjoint

2000 Mathematics Subject Classification: 03E35

Downloading the paper