We prove that for every positive integer m there is a finite point set \(\cal{P}\) in the plane such that no matter how \(\cal{P}\) is three-colored, there is always a disk containing exactly m points, all of the same color. This improves a result of Pach, Tardos and Tóth who proved the same for two colors. The main ingredient of the construction is a subconstruction whose points are in convex position. Namely, we show that for every positive integer m there is a finite point set \(\cal{P}\) in the plane in convex position such that no matter how \(\cal{P}\) is two-colored, there is always a disk containing exactly m points, all of the same color. We also prove that for unit disks no similar construction can work, and several other results.

我们证明对于每个正整数m在平面上都有一个有限点集\(\cal{P}\)使得无论\(\cal{P}\)如何是三色的，总有一个圆盘正好包含m个点，所有的颜色都相同。这改善了 Pach、Tardos 和 Tóth 的结果，他们证明了两种颜色的结果相同。构造的主要成分是一个子构造，其点在凸位置。即，我们证明对于每个正整数 m 在凸位置的平面中都有一个有限点集\(\cal{P}\)使得无论\(\cal{P}\)是双色的，总是有一个磁盘正好包含m点，都是一样的颜色。我们还证明了对于单位圆盘，没有类似的构造可以工作，以及其他几个结果。