THEORIES OF BALDWIN-SHI HYPERGRAPHS: THEIR ATOMIC MODELS AND REGULAR TYPES

Abstract

In [1], Baldwin and Shi studied the properties of generic structures built from certain Fraı̈ssé classes of weighted hypergraphs equipped with a notion of strong substructure. Here we focus on a particularly important class of such structures, where much stronger results are possible. We begin by fixing a finite relational language and a set of weights α. After constructing certain weighted hypergraphs with carefully chosen properties, we use these constructions to obtain an ∀∃-axiomatization for the theory of the generic, denoted by S α , and a quantifier elimination result for S α . These results, which extend those of Laskowski in [2] and Ikeda, Kikyo and Tsuboi in [3] are then used to study atomic and existentially closed models of S α , resulting in a necessary and sufficient condition on the weights that yields the existence of atomic models of thecorresponding theory. We then proceed to obtain the stability of S α and a characertization of non- forking, simplifying the proofs of some of these well known results (see [1], [4]) in the process. We identify conditions on α that guarantee that S α is non-trivial and prove that S α has the dimensional order property, a result that has only been established under certain additional hypothesis (see [5], [2]). Restricting ourselves to the case where the weights are all rational (excluding, what is essentially a single exception), we characterize the countable models up to isomorphism and show that they form an elementary chain of order type ω +

We also characterize the regular types of S α and explore the corresponding

pregeometries. We answer a question of Pillay in [6] by providing examples of pseudofinite stable theories with non-locally modular regular types. We conclude by studying the aforementioned exception (characterized by hav- ing trivial forking) and extending some of the results to countably infinite languages.