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arXiv · 2601.04417

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

Patrick Lutz
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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.

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Patrick Lutz. 2026-01-07. A Borel graphable equivalence relation with no Borel graphing of diameter two. https://arxiv.org/abs/2601.04417

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