On model-theoretic tree properties

TL;DR

This paper investigates model-theoretic tree properties, refining existing theorems, characterizing properties like $ ext{TP}_1$, and applying these insights to classify certain theories as $ ext{NSOP}_1$.

Contribution

It provides a quantitative refinement of Shelah's theorem for countable theories, shows $ ext{TP}_1$ is witnessed by a single-variable formula, and characterizes $ ext{NSOP}_1$ via independent amalgamation.

Findings

Refined Shelah's theorem for countable theories.

Proved $ ext{TP}_1$ is witnessed by a single-variable formula.

Characterized $ ext{NSOP}_1$ using independent amalgamation.

Abstract

We study model theoretic tree properties ($\text{TP}, \text{TP}_1, \text{TP}_2$) and their associated cardinal invariants ($\kappa_{\text{cdt}}, \kappa_{\text{sct}}, \kappa_{\text{inp}}$, respectively). In particular, we obtain a quantitative refinement of Shelah's theorem ($\text{TP} \Rightarrow \text{TP}_1 \lor \text{TP}_2$) for countable theories, show that $\text{TP}_1$ is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak $k-\text{TP}_1$ is equivalent to $\text{TP}_1$ (answering a question of Kim and Kim). Besides, we give a characterization of $\text{NSOP}_1$ via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are indeed $\text{NSOP}_1$.