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.

翻译：我们证明,对于任何平面的 C 体, 都有正整数 m, 其属性是, 任何定点P 都可以是三色的, 以至于没有 C 的译文包含至少 m 点P, 全部都是相同的颜色。 作为证据的一部分, 我们显示了 Erd\ H { { o} s- Sands- Sauer- Woodrow 的猜想。 令人惊讶的是, 证据还依赖于 光学猜想的两维情况 。