May 14
Corey Switzer, University of Vienna
Tight Maximal Eventually Different Families

Maximal almost disjoint (MAD) families and their relatives have been an important area of combinatorial and descriptive set theory since at least the 60s. In this talk I will discuss some relatives of MAD families, focussing on eventually different families of functions $f:\omega \to \omega$ and eventually different sets of permutations $p \in S(\omega)$. In the context of MAD families it has been fruitful to consider various strengthenings of the maximality condition to obtain several flavors of 'strongly' MAD families. One such strengthening that has proved useful in recent literature is that of tightness. Tight MAD families are Cohen indestructible and come with a properness preservation theorem making them nice to work with in iterated forcing contexts.

I will introduce a version of tightness for maximal eventually different families of functions $f:\omega \to \omega$ and maximal eventually different families of permutations $p \in S(\omega)$ respectively. These tight eventually different families share a lot of the nice, forcing theoretic properties of tight MAD families. Using them, I will construct explicit witnesses to $\mathfrak{a}_e= \mathfrak{a}_p = \aleph_1$ in many known models of set theory where this equality was either not known or only known by less constructive means. Working over $L$ we can moreover have the witnesses be $\Pi^1_1$ which is optimal for objects of size $\aleph_1$ in models where ${\rm CH}$ fails. These results simultaneously strengthen several known results on the existence of definable maximal sets of reals which are indestructible for various definable forcing notions. This is joint work with Vera Fischer.