MAD families and definability in various settings

An almost disjoint (AD) family is a family of infinite subsets of natural numbers that satisfies that the intersection of any two distinct members of the family is finite. We say that an almost disjoint family is maximal (MAD) if it is maximal under $\subseteq$ among almost disjoint families.

Infinite MAD families exists as a consequence of Zorns' lemma. The question of how definable such families can be in terms of topology have been a field of study since the 1960's, when Mathias proved that infinite MAD families cannot be analytic.

I will give a short introduction to MAD families and also generalize the notion to $\mathcal{I}$-MAD families where $\mathcal{I}$ is any Borel ideal on $\omega$. I will present recent results concerning the definability of such families in various settings. This is joint work with David Schrittesser and Asger Törnquist.