Abstract

A geometric graph is a graph drawn in the plane such that its edges are closed line seg-ments and no 3 vertices are collinear. We settle an old question of Avital, Hanani, Erdos, Kupitz and Perles by showing that every geometric graph with n vertices and m> k4n edges cent ains 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 them, or we can find an at most 2. 105 ks element subset of V meeting all rectangles. This im-proves a result of Ding, Seymour and Winkler. Both proofs are based on Dilworth’s theorem on

Citations