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 developed in [BL12], we interpret $⟨{\mathbb{Q}}^{+},<⟩$ in the $⟨{\mathcal{K}}_0^{+},{\le }^{\ast }⟩$-generic and prove that this structure is not pseudofinite. This provides a negative answer to the question posed in [EW09] (Question 2.6). This result, in fact, unfolds another aspect of complexity of this structure, along with undecidability and strict order property proved in [EW09] and [Bl12]. On the other hand, when $R$ is binary, it can be shown that the $⟨{\mathcal{K}}_0^{+},{\le }^{\ast }⟩$-generic is decidable and pseudofinite.

to appear in Notre Dame Journal of Formal Logic