Three-chromatic geometric hypergraphs

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 in such a way that no translate of C contains m points of P (or more), all of the same color. As a part of the proof, we show a strengthening of the Erdős–Sands–Sauer–Woodrow conjecture. Surprisingly, the proof also relies on the two-dimensional case of the Illumination Conjecture. The extended abstract of this paper already appeared in the proceedings of SoCG ’22.