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Pseudofiniteness of the Farey Graph

Connor Martinez Lockhart

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.

Pseudofiniteness of the Farey Graph

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.
Paper Structure (9 sections, 12 theorems, 1 equation)

This paper contains 9 sections, 12 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. The Farey Graph
  3. Graph Theoretic Preliminaries
  4. Graph Embeddings
  5. Main Results
  6. Verification of $\psi_n$
  7. Verification of $\phi_0$
  8. Pseudofiniteness
  9. Bibliography

Key Result

Proposition 1

A vertex is removable if and only if its neighborhood $N[v]$ is a clique of size at most 3. In particular, if $G$ is $K_4$-free, a vertex is removable if and only if its neighborhood is a clique.

Theorems & Definitions (35)

  • Definition 1: Pseudofinite Structure
  • Definition 2: Pseudofinite Theory
  • Definition 3: Classical Farey Graph
  • Definition 4: Combinatorial Construction
  • Definition 5: Vertex Connectivity
  • Definition 6
  • Definition 7
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 25 more