Three-chromatic geometric hypergraphs

TL;DR

This paper proves a three-coloring property for finite point sets in the plane with respect to convex bodies, linking geometric hypergraph coloring to longstanding conjectures and demonstrating a new combinatorial geometric result.

Contribution

It establishes a universal three-coloring result for planar point sets avoiding large monochromatic translates of any convex body, and strengthens the Erdős–Sands–Sauer–Woodrow conjecture.

Findings

Existence of a universal integer m for three-coloring point sets

Connection between hypergraph coloring and the Illumination conjecture

Strengthening of the Erdős–Sands–Sauer–Woodrow conjecture

Abstract

We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erd\H{o}s-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.