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 Erdős-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 Károlyi et. al. [5], Kun and Lippner [7] and Valtr [11],[12]. A set of points is k-convex if it determines no triangle with more than k points inside. Supported by the project 1M0021620808 of the Ministry of Education of the Czech Republic.