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 p 1 , p 2 , … , p m according to the increasing x -coordinate. The set Y is open in X if there is no point of X above the polygonal line p 1 , p 2 , … , p m . Valtr [P. Valtr, Open caps and cups in planar point sets, DCG (in press)] showed that for every two 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 recurrences for g ( k , l ) . This theorem implies results on empty polygons in k ′ -convex sets proved by Károlyi et al. [Gy. Károlyi, J. Pach, G. Tóth, A modular version of the Erdős–Szekeres theorem, Studia Sci. Math. Hungar. 38 (2001) 245–259], Kun and Lippner [G. Kun, G. Lippner, Large convex empty polygons in k -convex sets, Period. Math. Hungar. 46 (2003) 81–88] and Valtr [P. Valtr, A sufficient condition for the existence of large empty convex polygons, Discrete Comput. Geom. 28 (2002) 671–682; P. Valtr, Open caps and cups in planar point sets, DCG (in press)]. A set of points is k ′ -convex if it determines no triangle with more than k ′ points inside.