How to drive our families mad

Given a family F of pairwise almost disjoint (ad) sets on a countable set S, we study families F̃ of maximal almost disjoint (mad) sets extending F . We define a+(F) to be the minimal possible cardinality of F̃ \F for such F̃ and a+(κ) = max{a+(F) : | F | ≤ κ}. We show that all infinite cardinal less than or equal to the continuum c can be represented as a+(F) for some ad F (Theorem 21) and that the inequalities א1 = a < a (א1) = c (Corollary 19) and a = a(א1) < c (Theorem 20) are both consistent. We also give a several constructions of mad families with some additional properties.