Proper Coloring of Geometric Hypergraphs
Bal\'azs Keszegh, D\"om\"ot\"or P\'alv\"olgyi

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.
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