Disjoint $n$-amalgamation and pseudofinite countably categorical theories

TL;DR

This paper explores conditions under which certain countably categorical theories are pseudofinite, focusing on disjoint n-amalgamation, and demonstrates that specific non-simple theories like $T^*_{feq}$ and $T_{CPZ}$ are pseudofinite, including for the first time their NSOP$_1$ status.

Contribution

It establishes that theories with disjoint n-amalgamation for all n are pseudofinite and extends methods to non-simple theories, providing new examples and properties.

Findings

Countably categorical theories with disjoint n-amalgamation are pseudofinite.

Certain non-simple theories like $T^*_{feq}$ and $T_{CPZ}$ are pseudofinite.

The theory $T_{CPZ}$ is shown to be NSOP$_1$ for the first time.

Abstract

Disjoint $n$-amalgamation is a condition on a complete first-order theory specifying that certain locally consistent families of types are also globally consistent. In this paper, we show that if a countably categorical theory $T$ admits an expansion with disjoint $n$-amalgamation for all $n$, then $T$ is pseudofinite. All theories which admit an expansion with disjoint $n$-amalgamation for all $n$ are simple, but the method can be extended, using filtrations of Fra\"iss\'e classes, to show that certain non-simple theories are pseudofinite. As case studies, we examine two generic theories of equivalence relations, $T^*_{\text{feq}}$ and $T_{\text{CPZ}}$, and show that both are pseudofinite. The theories $T^*_{\text{feq}}$ and $T_{\text{CPZ}}$ are not simple, but they are NSOP$_1$. This is established here for $T_{\text{CPZ}}$ for the first time.