Holes in 2-convex point sets

Abstract Let S be a set of n points in the plane in general position (no three points from S are collinear). For a positive integer k , a k-hole in S is a convex polygon with k vertices from S and no points of S in its interior. For a positive integer l , a simple polygon P is l-convex if no straight line intersects the interior of P in more than l connected components. A point set S is l-convex if there exists an l -convex polygonization of S . Considering a typical Erdős–Szekeres-type problem, we show that every 2-convex point set of size n contains an Ω ( log ⁡ n ) -hole. In comparison, it is well known that there exist arbitrarily large point sets in general position with no 7-hole. Further, we show that our bound is tight by constructing 2-convex point sets in which every hole has size O ( log ⁡ n ) .