**Sakaé Fuchino, Stefan Geschke and
Lajos Soukup**

Given a family of pairwise almost disjoint (ad) sets on a countable set , we study families of maximal almost disjoint (mad) sets extending .

We define to be the minimal possible cardinality of for such and . We show that all infinite cardinal less than or equal to the continuum can be represented as for some ad and that the inequalities and 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