A Borel graphable equivalence relation with no Borel graphing of diameter two
Logic
2026-01-09 v1
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.
Cite
@article{arxiv.2601.04417,
title = {A Borel graphable equivalence relation with no Borel graphing of diameter two},
author = {Patrick Lutz},
journal= {arXiv preprint arXiv:2601.04417},
year = {2026}
}
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8 pages