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arXiv · 2011.12187

Realizing an m-uniform four-chromatic hypergraph with disks

Gábor Damásdi·Pálvölgyi Dömötör
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Abstract

We prove that for every $m$ there is a finite point set $\mathcal{P}$ in the plane such that no matter how $\mathcal{P}$ 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 T\'oth 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 $\mathcal{P}$ in the plane in convex position such that no matter how $\mathcal{P}$ 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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Gábor Damásdi, Pálvölgyi Dömötör. 2020-11-24. Realizing an m-uniform four-chromatic hypergraph with disks. https://arxiv.org/abs/2011.12187

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