Disjoint n-Amalgamation and Pseudofinite Countably Categorical Theories

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 article, 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 Fraïssé classes, to show that certain nonsimple theories are pseudofinite. As case studies, we examine two generic theories of equivalence relations, $T_{\mathrm{feq}}^{\ast }$ and $T_{\mathrm{CPZ}}$, and show that both are pseudofinite. The theories $T_{\mathrm{feq}}^{\ast }$ and $T_{\mathrm{CPZ}}$ are not simple, but they have NSOP${}_1$. This is established here for $T_{\mathrm{CPZ}}$ for the first time.