Proper Coloring of Geometric Hypergraphs

TL;DR

This paper investigates the 3-coloring problem for geometric hypergraphs, focusing on pseudo-disks and convex polygons, and explores extensions to higher dimensions.

Contribution

It proves the existence of such colorings for homothetic convex polygons and discusses the problem's extension to higher-dimensional spaces.

Findings

Existence of a finite m for convex polygons

Partial results for pseudo-disk families

Extension considerations to higher dimensions

Abstract

We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then such an m exists. We prove this in the special case when F is the family of all homothetic copies of a given convex polygon. We also study the problem in higher dimensions.