We prove that for every m there is a finite point set P in the plane such that no matter how 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 m there is a finite point set P in the plane in convex position such that no matter how 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.
We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with 2k k steps with generalized rectangles there is a row or a column in the diagram that is used by at least k + 1 rectangles, and prove that this is best-possible.
We prove that for every m there is a finite point set P in the plane such that no matter how 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 m there is a finite point set P in the plane in convex position such that no matter how 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.
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