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$_{2}$.