Three-chromatic geometric hypergraphs

Damásdi, Gábor and Pálvölgyi, Dömötör (2025) Three-chromatic geometric hypergraphs. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. ISSN 1435-9855 (In Press)

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

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