Three-chromatic geometric hypergraphs
Combinatorics
2026-01-21 v2 Discrete Mathematics
Abstract
We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erd\H{o}s-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.
Cite
@article{arxiv.2112.01820,
title = {Three-chromatic geometric hypergraphs},
author = {Gábor Damásdi and Dömötör Pálvölgyi},
journal= {arXiv preprint arXiv:2112.01820},
year = {2026}
}
Comments
In the revised version we have removed Appendix B, which contained an incorrect proof of a footnote