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

TL;DR

This work analyzes NTP$_{2}$-style phenomena by introducing $NTP_{2}$-smooth Keisler measures and exploring distal-like behavior within NTP$_{2}$. It proves that every measure over a model in an $NTP_{2}$ theory extends to an $NTP_{2}$-smooth one and proposes a candidate $NTP_{2}$-distality based on the smoothness of average measures over indiscernibles. It also establishes a finite alternation theorem under $oldsymbol{f}$-resilience, showing a strong tameness property for dividing formulas in a broad subclass. Finally, it proves a singular local character result in NIP and discusses potential extensions to $NTP_{2}$, highlighting open questions and interplay between NTP$_{2}$ and simple theories. These results provide new tools for understanding the boundary between NIP-like stability and instability in NTP$_{2}$-rich theories.

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$_{2}$.

Theorems & Definitions (42)

• Definition 2.1
• Definition 2.2
• Definition 3.1
• Remark 3.3
• Remark 3.4
• Definition 3.5
• Remark 3.6
• Proposition 3.8
• proof
• Theorem 3.9