Unfriendly partitions of locally finite Borel graphs
Abstract
We answer in the negative the question of Thomas, recorded by Conley, Conley--Marks--Unger, and Conley--Tamuz, of whether every locally finite Borel graph admits a Borel unfriendly partition. Our counterexample has unbounded degree and is closed on a zero-dimensional Polish space; its connectedness relation is hyperfinite, and its components are bipartite and one-ended. Every unfriendly colouring is proper. Together with a parity obstruction, this rigidity rules out Baire measurable colourings that are unfriendly on a comeager set, and measurable colourings that are unfriendly almost everywhere for a quasi-invariant probability of finite average degree. In the positive direction, a Borel graph of maximum degree at most four admits a Borel unfriendly colouring whenever each component contains a cycle or a vertex of degree at most two. This reduces the Borel problem in maximum degree three to cubic forests and, with a theorem of Conley--Marks--Unger, gives Baire measurable unfriendly colourings for all Borel graphs of maximum degree at most four.
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BibTeX
@misc{pelayogomez2026unfriendly,
title = {{Unfriendly partitions of locally finite Borel graphs}},
author = {José de Jesús Pelayo-Gómez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11919},
eprint = {2609.11919},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11919},
}
RIS
TY - UNPB
TI - Unfriendly partitions of locally finite Borel graphs
AU - José de Jesús Pelayo-Gómez
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11919
UR - https://arxiv.org/abs/2609.11919
AB - We answer in the negative the question of Thomas, recorded by Conley, Conley--Marks--Unger, and Conley--Tamuz, of whether every locally finite Borel graph admits a Borel unfriendly partition. Our counterexample has unbounded degree and is closed on a zero-dimensional Polish space; its connectedness relation is hyperfinite, and its components are bipartite and one-ended. Every unfriendly colouring is proper. Together with a parity obstruction, this rigidity rules out Baire measurable colourings that are unfriendly on a comeager set, and measurable colourings that are unfriendly almost everywhere for a quasi-invariant probability of finite average degree. In the positive direction, a Borel graph of maximum degree at most four admits a Borel unfriendly colouring whenever each component contains a cycle or a vertex of degree at most two. This reduces the Borel problem in maximum degree three to cubic forests and, with a theorem of Conley--Marks--Unger, gives Baire measurable unfriendly colourings for all Borel graphs of maximum degree at most four.
ER -