Mad families constructed from perfect almost disjoint families

Abstract

We prove the consistency of $\mathfrak{b} > \aleph_1$ together with the existence of a $\Pi^1_1$-definable mad family, answering a question posed by Friedman and Zdomskyy in [7, Question 16]. For the proof we construct a mad family in $L$ which is an $\aleph_1$-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Borel almost-disjointness number $\mathfrak{a}_B$, defined as the least number of Borel a.d. sets whose union is a mad family. Our proof yields the consistency of $\mathfrak{a}_B < \mathfrak{b}$ (and hence, $\mathfrak{a}_B < \mathfrak{a}$).