Discrete Comput Geom 4:245-251 (1989) Ih~crete d, C,,ml~llat~mal eometrv"~. 1989 Spr~nger-Verlag New York lnc ~" Summary: Discrete Comput Geom 4:245-251 (1989) Ih~crete d, C,,ml~llat~mal eometrv"~. 1989 Spr~nger-Verlag New York lnc ~" The Maximum Size of a Convex Polygon in a Restricted Set of Points in the Plane* N. Alon, r M. Katchalski, 2 and W. R. Pulleyblank 3 Department of Mathematics, Tel AvJv University. Tel Aviv, Israel 2Department of Mathematics, Technion-lsrael Institute of Technology, Haifa, Israel 3Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada Abstract. Assume we have k points in general position in the plane such that the ratio between the maximum distance of any pair of points to the minimum distance of any pair of points is at most crx/k, for some positive constant a. We show that there exist at least flk 1/4of these points which are the vertices of a convex polygon, for some positive constant /3 =/3(a). On the other hand, we show that for every fixed e>0, if k>k(e), then there is a set of k points in the plane for which the above ratio is at most 4~, which does not contain a convex polygon of more than k 1/3+~vertices. 1. Introduction For any positive integer n, let f(n) be the smallest integer such that from every set off(n) points in general position in the plane (i.e., no three are on a line), it is always possible to select n points which are the vertices of a convex n-gon. Collections: Mathematics