KGRC Research Seminar – 2017-06-22 at 4pm.

**Speaker:** David Schrittesser (University of Copenhagen, Denmark)

**Abstract: **This talk is about two results on mad families (dating from this year): Firstly, in joint work with Asger Törnquist and Karen Bakke Haga, we link madness of certain definable sets to forcing and use this to show that under the Axiom of Projective Determinacy there are no projective mad families. Moreover, the results generalize: we may replace “being almost disjoint” by “being $J$-disjoint”, for certain ideals $J$ on the natural numbers including, e.g., Fin $\times$ Fin. The other result is an improvement of Horowitz and Shelah’s construction of a Borel maximal eventually different family of functions. We obtain a closed such family, and the result even generalizes to certain compact spaces.