Realizing an m-uniform four-chromatic hypergraph with disks

TL;DR

This paper demonstrates the existence of finite point sets in the plane that, regardless of three-coloring, always contain a disk with exactly m points of the same color, extending previous two-color results.

Contribution

It constructs point sets in convex position that guarantee monochromatic disks of size m under any two-coloring, and extends this to three colors, also analyzing limitations with unit disks.

Findings

Existence of point sets with monochromatic disks of size m for any m

Extension of previous two-color results to three colors

Limitations of the construction with unit disks

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.