 
Summary: Discrete Comput Geom 4:245251 (1989) Ih~crete d, C,,ml~llat~mal
eometrv"~. 1989 Spr~ngerVerlag 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, Technionlsrael 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 ngon.
