Some geometric applications of Dilworth’s theorem
A geometric graph is a graph drawn in the plane such that its edges are closed line segments and no three vertices are collinear. We settle an old question of Avital, Hanani, Erdős, Kupitz, and Perles by showing that every geometric graph withn vertices andm>k4n edges containsk+1 pairwise disjoint edges. We also prove that, given a set of pointsV and a set of axis-parallel rectangles in the plane, then either there arek+1 rectangles such that no point ofV belongs to more than one of them, or we can find an at most 2·105k8 element subset ofV meeting all rectangles. This improves a result of Ding, Seymour, and Winkler. Both proofs are based on Dilworth’s theorem on partially ordered sets.
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