Some geometric applications of Dilworth's theorem

J. Pach, J. Törőcsik
1994 Discrete & Computational Geometry  
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, Erd~s, Kupitz, and Perles by showing that every geometric graph with n vertices and m > k4n edges contains k + 1 pairwise disjoint edges. We also prove that, given a set of points V and a set of axis-parallel rectangles in the plane, then either there are k + 1 rectangles such that no point of V belongs to more than one of
more » ... them, or we can find an at most 2" 105k a element subset of V meeting all rectangles. This improves a result of Ding, Seymour, and Winkler. Both proofs are based on Dilworth's theorem on partially ordered sets.
doi:10.1007/bf02574361 fatcat:syigyh5vujevfdv4proqmrp67i