A Borel graphable equivalence relation with no Borel graphing of diameter two

TL;DR

This paper constructs a specific Borel graphable equivalence relation that cannot be represented with a Borel graph of diameter less than 3, highlighting limitations in Borel graphings.

Contribution

It demonstrates the existence of a Borel graphable equivalence relation with a minimal diameter of at least 3, answering a previously open question.

Findings

Existence of a Borel graphable equivalence relation with no Borel graphing of diameter less than 3

Constructs a Borel graph with diameter at most 4

Introduces a technical lemma on computability-theoretic genericity

Abstract

We answer a question of Arant, Kechris and Lutz by showing that there is a Borel graphable equivalence relation with no Borel graphing of diameter less than 3. More specifically, we prove that there is an equivalence relation with a Borel graphing of diameter at most 4 but no Borel graphing of diameter less than 3. Our proof relies on a technical lemma about computability-theoretic genericity, which may have other applications.