Some NIP-like phenomena in NTP${}_2$

Abstract

We introduce the notion of an NTP${}_2$-smooth measure and prove that they exist assuming NTP${}_2$. Using this, we propose a notion of distality in NTP${}_2$ that unfortunately does not intersect simple theories trivially. We then prove a finite alternation theorem for a subclass of NTP${}_2$ that contains resilient theories. In the last section we prove that under NIP, any type over a model of singular size is finitely satisfiable in a smaller model, and ask if a parallel result (with non-forking replacing finite satisfiability) holds in NTP${}_2$.