The measure half of the 2n+1 problem
Abstract
Let F0,…,Fn−1 be Borel functions on a standard Borel space X and let GF be the graph they generate. We prove that for every Borel probability measure μ on X there are a forward-invariant μ-conull Borel set A ⊆ X and a proper Borel (2n+1)-coloring of the restriction of GF to A; consequently χM(GF) ≤ 2n+1 in the total sense of Kechris and Marks. This answers positively, for every n and with the optimal constant, the measure half of Problem 5.14 of the survey "Descriptive graph combinatorics" of Kechris and Marks -- the measurable version of a question raised by Kechris, Solecki, and Todorcevic (Adv. Math. 141, 1999). No invariance, no local finiteness, and no finiteness of the Borel chromatic number are assumed. The argument is elementary and specific to measure; the corresponding Borel problem and the Baire-measurable half remain open.
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BibTeX
@misc{gomez2026measure,
title = {{The measure half of the $2n+1$ problem}},
author = {José de Jesús Pelayo Gómez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19038},
eprint = {2609.19038},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19038},
}
RIS
TY - UNPB
TI - The measure half of the 2n+1 problem
AU - José de Jesús Pelayo Gómez
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.19038
UR - https://arxiv.org/abs/2609.19038
AB - Let F_0,…,F_n−1 be Borel functions on a standard Borel space X and let G_F be the graph they generate. We prove that for every Borel probability measure μ on X there are a forward-invariant μ-conull Borel set A ⊆ X and a proper Borel (2n+1)-coloring of the restriction of G_F to A; consequently χ_M(G_F) ≤ 2n+1 in the total sense of Kechris and Marks. This answers positively, for every n and with the optimal constant, the measure half of Problem 5.14 of the survey "Descriptive graph combinatorics" of Kechris and Marks -- the measurable version of a question raised by Kechris, Solecki, and Todorcevic (Adv. Math. 141, 1999). No invariance, no local finiteness, and no finiteness of the Borel chromatic number are assumed. The argument is elementary and specific to measure; the corresponding Borel problem and the Baire-measurable half remain open.
ER -