A note on NSOP$_{1}$ in one variable

TL;DR

This paper shows that to determine if a theory is NSOP$_{1}$, it is enough to verify that no single-variable formula exhibits SOP$_{1}$, simplifying the classification process.

Contribution

It introduces a criterion reducing the complexity of checking NSOP$_{1}$ by focusing on formulas with one free variable.

Findings

NSOP$_{1}$ can be characterized by single-variable formulas.

Simplifies the process of classifying theories as NSOP$_{1}$.

Provides a new approach to understanding SOP$_{1}$ in model theory.

Abstract

We prove that, in order to establish that a theory is NSOP$_{1}$, it suffices to show that no formula in a single free variable has SOP$_{1}$.