Introduction to algorithms
Ramsey theory (2nd ed.)
Applications of the crossing number
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
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A geometric graph is a graph drawn in the plane such that its edgesare closed line segments and no 3 vertices are collinear. We settle anold question of Avital, Hanani, Erdo&huml:s, Kupitz and Perles byshowing that every geometric graph withn vertices and m k4n edges containsk+1 pairwise disjoint edges. We alsoprove that, given a set of points V and a set of axis-parallelrectangles 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 most2˙105k8 element subset of V meeting allrectangles. This improves a result of Ding, Seymour and Winkler. Bothproofs are based on Dilworth's theorem on partially ordered sets.