A simple proof for open cups and caps

Let X be a set of points in general position in the plane. General position means that no three points lie on a line and no two points have the same x-coordinate. Y X is a cup, resp. cap, if the points of Y lie on the graph of a convex, resp. concave function. Denote the points of Y by p1, p2, . . . , pm according to the increasing x-coordinate. The set Y is open in X if there is no point of X above the polygonal line p1, p2, . . . , pm. Valtr [12] showed that for every positive integers k and l there exists a positive integer g(k, l) such that any g(k, l)point set in the plane in general position contains an open k-cup or an open l-cap. This is a generalization of the Erdos-Szekeres theorem on cups and caps. We show a simple proof for this theorem and we also show better recurreces for g(k, l). This theorem implies results on empty polygons in k -convex sets proved by K

• Type: Journal
• Year: 2008
• Where: EJC
• Authors: Jakub Cerný
• Keywords: EJC 2007, EJC 2008, General Position, Information Technology, Polygonal Line P1, Positive Integer