Pseudofiniteness in Hrushovski Constructions

TL;DR

The paper studies pseudofiniteness in Fraïssé-Hrushovski constructions with a single relation $R$ whose arity is either $2$ or $3$. It shows a sharp arity-based dichotomy: the $\langle\mathcal{K}^{+}_{0},\leq^{*}\rangle$-generic is not pseudofinite in the ternary case by interpreting $(\mathbb{Q}^{+},<)$ inside it, while in the binary case the theory is decidable and pseudofinite, with a complete axiomatization $UNIV$ and a corresponding acyclic-graph interpretation. The methodology hinges on definable coding tools $R^{k}$ that code finite subsets and minimal pairs, enabling interpretability arguments in the ternary setting. These results clarify the boundary between pseudofiniteness and decidability for Hrushovski-type generics and connect to prior work on instability, the strict order property, and related model-theoretic behavior.

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 $ \langle\mathbb{Q}^{+},<\rangle $ in the $ \langle\mathcal{K}^{+}_{0},\leq^{*}\rangle $-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 $ \langle\mathcal{K}^{+}_{0},\leq^{*}\rangle $-generic is decidable and pseudofinite.

Theorems & Definitions (24)

• Definition 1.1
• Definition 1.2
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Remark 1.8
• Lemma 2.1
• proof
• Lemma 2.2
• Theorem 2.3