Mathias showed in 1969 that no maximal almost disjoint family can be analytic; in 1989 Arnold Miller showed under \(V=L\) there exists a coanalytic mad family, and in 2010 Friedman and Zdomskyy constructed a model in which the continuum is of size \(\aleph_2\) and there exists a \(\Pi_2^1\) tight mad family of size \(\aleph_2\). In this talk I will introduce the Friedman-Zdomskyy forcing and its preservation properties, outlining the construction of a model in which \(2^{\aleph_0} = \aleph_2\) and there exists a \(\Pi_1^1\) tight mad family of size \(\aleph_1\) as well as a \(\Pi_2^1\) tight mad family of size \(\aleph_2\); each of these projective definitions is of lowest possible descriptive complexity. Namely this involves a notion of preservation of mad families given by Guzman, Hrušák, and Tellez, and if time permits I will talk about this property for the case of a certain forcing notion given by Shelah in 1984.
This is joint work with Vera Fischer.