COUNTABLE MODELS OF THE THEORIES OF BALDWIN-SHI HYPERGRAPHS AND THEIR REGULAR TYPES

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 used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain {M-beta: beta <= omega} of countable models of the theory of a fixed Baldwin-Shi hypergraph with M-beta <= M-gamma if and only if the dimension of M-beta is at most the dimension of M-gamma and that each countable model is isomorphic to some 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 a large body of work, we use these structures to give an example of a pseudofinite, omega-stable theory with a nonlocally modular regular type, answering a question of Pillay in [11].