Some NIP-like phenomena in NTP$_{2}$

TL;DR

This paper explores properties of NTP$_{2}$ theories, introducing NTP$_{2}$-smooth measures, proposing a notion of distality, and establishing a finite alternation theorem, thereby advancing understanding of NTP$_{2}$ model theory.

Contribution

It introduces NTP$_{2}$-smooth measures, proposes a new notion of distality in NTP$_{2}$, and proves a finite alternation theorem for a subclass of NTP$_{2}$ theories.

Findings

Existence of NTP$_{2}$-smooth measures assuming NTP$_{2}$.

A finite alternation theorem for resilient theories within NTP$_{2}$.

Under NIP, types over singular models are finitely satisfiable in smaller models.

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$_{2}$.