Three-chromatic geometric hypergraphs

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 in such a way that no translate of C contains m points of  P (or more), all of the same color. As a part of the proof, we show a strengthening of t...

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Bibliographic Details
Published inJournal of the European Mathematical Society : JEMS
Main Authors Damásdi, Gábor, Pálvölgyi, Dömötör
Format Journal Article
LanguageEnglish
Published 28.08.2024
Online AccessGet full text
ISSN1435-9855
1435-9863
1435-9863
DOI10.4171/jems/1516

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Summary: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 in such a way that no translate of C contains m points of  P (or more), all of the same color. As a part of the proof, we show a strengthening of the Erdős–Sands–Sauer–Woodrow conjecture. Surprisingly, the proof also relies on the two-dimensional case of the Illumination Conjecture. The extended abstract of this paper already appeared in the proceedings of SoCG ’22.
ISSN:1435-9855
1435-9863
1435-9863
DOI:10.4171/jems/1516