On finite approximations of transitive graphs

TL;DR

This paper constructs a specific infinite, vertex-transitive graph that cannot be approximated by any finite local model for sufficiently large radii, addressing a question in graph theory.

Contribution

It provides an example of a transitive graph with no perfect finite local approximations for large radii, answering a previously open question.

Findings

Existence of a vertex-transitive, unimodular, locally finite graph without finite local models for large r

Counterexample to the assumption that all transitive graphs can be approximated locally

Clarification of limitations in finite approximations of infinite graphs

Abstract

In this note we answer a question of Johannes Carmesin, which was circulated at the Oberwolfach Workshop on "Graph Theory" in January 2025. We provide a unimodular, locally finite, and vertex-transitive graph without any perfect finite $r$-local model for $r \in \mathbb N$ large enough.