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 Erdős-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.

翻译：我们证明，对于任意平面凸体C，存在一个正整数m，使得平面上任意有限点集P可被三染色，且不存在C的平移包含至少m个P中同色点。作为证明的一部分，我们给出了Erdős-Sands-Sauer-Woodrow猜想的一个强化形式。令人惊讶的是，该证明还依赖于照明猜想的二维情形。