Pseudofiniteness in Hrushovski Constructions

Abstract

In a relational language consisting of a single relation $R$, we investigate pseudofiniteness of certain Hrushovski constructions obtained via predimension functions. It is notable that the arity of the relation $R$ plays a crucial role in this context. When $R$ is ternary, by extending the methods recently developed by Brody and Laskowski, we interpret $\langle {\mathbb{Q}}^{+},<\rangle$ in the $\langle {\mathcal{K}}^{+},{\le }^{\ast }\rangle$-generic and prove that this structure is not pseudofinite. This provides a negative answer to the question posed in an earlier work by Evans and Wong. This result, in fact, unfolds another aspect of complexity of this structure, along with undecidability and the strict order property proved in the mentioned earlier works. On the other hand, when $R$ is binary, it can be shown that the $\langle {\mathcal{K}}^{+},{\le }^{\ast }\rangle$-generic is decidable and pseudofinite.