Large empty convex polygons in k-convex sets

Abstract

A point set $P$ is k-convex if there are at most k points of $P$ in any triangle having its vertices in $P$. Károlyi, Pach and Tóth [6] showed that if a 1-convex set has sufficiently many points, then it contains an arbitrarily large emtpy convex polygon. They also constructed exponentially large 1-convex sets that contain no empty convex n-gons. Here we shall give an exponential upper bound to the number of points needed. Valtr [8] proved a similar result for k-convex sets. In this paper we improve his upper bound and give an elementary proof of the statement.