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