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Realizing an m-Uniform Four-Chromatic Hypergraph with Disks

Combinatorica volume 42, pages 1027–1048 (2022)Cite this article

Abstract

We prove that for every positive integer m there is a finite point set \(\cal{P}\) in the plane such that no matter how \(\cal{P}\) is three-colored, there is always a disk containing exactly m points, all of the same color. This improves a result of Pach, Tardos and Tóth who proved the same for two colors. The main ingredient of the construction is a subconstruction whose points are in convex position. Namely, we show that for every positive integer m there is a finite point set \(\cal{P}\) in the plane in convex position such that no matter how \(\cal{P}\) is two-colored, there is always a disk containing exactly m points, all of the same color. We also prove that for unit disks no similar construction can work, and several other results.

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Acknowledgment

We would like to thank Balázs Keszegh for useful discussions and for reading a draft of this manuscript, and the anonymous reviewers of the conference version for their suggestions.

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Authors and Affiliations

  1. MTA-ELTE Lendület Combinatorial, Geometry Research Group, ELTE Eötvös Loránd University, Budapest, Hungary

    Gábor Damásdi & Dömötör Pálvölgyi

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  1. Gábor Damásdi
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  2. Dömötör Pálvölgyi
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Correspondence to Dömötör Pálvölgyi.

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Damásdi, G., Pálvölgyi, D. Realizing an m-Uniform Four-Chromatic Hypergraph with Disks. Combinatorica 42 (Suppl 1), 1027–1048 (2022). https://doi.org/10.1007/s00493-021-4846-5

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  • DOI: https://doi.org/10.1007/s00493-021-4846-5

Mathematics Subject Classification (2010)

  • 05C15