Pseudofiniteness of the Farey Graph

TL;DR

This paper proves the theory of the Farey graph is pseudofinite by constructing finite models satisfying its axioms, revealing its model-theoretic properties and connections to surface triangulations.

Contribution

It demonstrates the pseudofiniteness of the Farey graph's theory through finite structure constructions and explores its relation to surface triangulations.

Findings

The Farey graph's theory is pseudofinite.

Finite models can satisfy the axioms on higher genus surfaces.

No finite planar graph satisfies the axioms for large substructures.

Abstract

We prove that the theory of the Farey graph is pseudofinite by constructing a sequence of finite structures that satisfy increasingly large subsets of its first-order axiomatization. This graph is an important object in the study of curve graphs, and its model-theoretic properties have been explored in the broader context of curve graphs of surfaces in arXiv:2008.10490 The theory of the Farey graph was recently axiomatized by Tent and Mohammadi in arXiv:2503.02121 We show that while no finite planar graph can satisfy these axioms for sufficiently large substructures, they can be satisfied by triangulations densely embedded on orientable surfaces of higher genus. By applying a result of Archdeacon, Hartsfield, and Little on the existence of triangulations with representativity and connectedness, we establish that every finite subset of the theory of the Farey graph has a finite model as desired.