Countable models of the theories of Baldwin-Shi hypergraphs and their regular types

Abstract

We continue the study of the theories of Baldwin-Shi hypergraphs from $[5]$. Restricting our attention to when the rank $\delta$ is rational valued, we show that each countable model of the theory of a given Baldwin-Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class of finite structures with the inherited notion of strong substructure. We introduce a notion of dimension for a model and show that there is a an elementary chain $\{\mathfrak{M}_{\beta}:\beta<\omega+1\}$ of countable models of the theory of a fixed Baldwin-Shi hypergraph with $\mathfrak{M}_{\beta}\preccurlyeq\mathfrak{M}_\gamma$ if and only if the dimension of $\mathfrak{M}_\beta$ is at most the dimension of $\mathfrak{M}_\gamma$ and that each countable model is isomorphic to some $\mathfrak{M}_\beta$. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on the work of Brody and Laskowski, we use these structures to give an example of a pseudofinite, $\omega$-stable theory with a non-locally modular regular type, answering a question of Pillay in $[9]$.