On k-gons and k-holes in point sets ?
We consider a variation of the classical Erd?sSzekeres problems on the existence and number of convex k-gons and k-holes (empty k-gons) in a set of n points in the plane. Allowing the k-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any k and sufficiently large n, we give a quadratic lower bound for the number of k-holes, and show that this number is maximized by sets in convex position.
k-gons k-holes erd?sszekeres type problems empty k-gons