Realizing an m-uniform four-chromatic hypergraph with disks
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We prove that for every m there is a finite point set 𝒫 in the plane such that no matter how 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 m there is a finite point set 𝒫 in the plane in convex position such that no matter how 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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