Some geometric applications of Dilworth's theorem

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 with<?Pub Fmt italic>n<?Pub Fmt /italic> vertices and <?Pub Fmt italic>m> k<supscrpt>4</supscrpt>n<?Pub Fmt /italic> edges contains<?Pub Fmt italic>k<?Pub Fmt /italic>+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 are<?Pub Fmt italic>k<?Pub Fmt /italic>+1 rectangles such that no point ofV belongs to more than one of them, or we can find an at most<inline-equation><f>2˙10⁵k⁸</f></inline-equation> <?Pub Caret>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.