## 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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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