Realizing an m-Uniform Four-Chromatic Hypergraph with Disks

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.