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 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.