Three-Chromatic Geometric Hypergraphs

Authors Gábor Damásdi , Dömötör Pálvölgyi

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Author Details

Gábor Damásdi
  • MTA-ELTE Lendület Combinatorial Geometry Research Group, Dept. of Computer Science, ELTE Eötvös Loránd University, Budapest, Hungary
Dömötör Pálvölgyi
  • MTA-ELTE Lendület Combinatorial Geometry Research Group, Dept. of Computer Science, ELTE Eötvös Loránd University, Budapest, Hungary

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Gábor Damásdi and Dömötör Pálvölgyi. Three-Chromatic Geometric Hypergraphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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 Erdős-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Hypergraphs
  • Discrete geometry
  • Geometric hypergraph coloring
  • Decomposition of multiple coverings


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