Let k ≥ 4. A finite planar point set X is called a convex k-clustering if it is a disjoint union of k sets X 1 , . . . , X k of equal sizes such that x 1 x 2 · · · x k is a convex k-gon for each choice of x 1 ∈ X 1 , . . . , x k ∈ X k . Answering a question of Gil Kalai, we show that for every k ≥ 4 there are two constants c = c(k), c = c (k) such that the following holds. If X is a finite set of points in general position in the plane, then it has a subset X of size at most c such that X \X can be partitioned into at most c convex k-clusterings. The special case k = 4 was proved earlier by Pór. Our result strengthens the so-called positive fraction Erdős-Szekeres theorem proved by Bárány and Valtr. The proof gives reasonable estimates on c and c , and it works also in higher dimensions. We also improve the previous constants for the positive fraction Erdős-Szekeres theorem obtained by Pach and Solymosi.

Let P be a set of points in general position in the plane. We say that P is k-convex if no triangle determined by P contains more than k points of P in the interior. We say that a subset A of P in convex position forms an empty polygon (in P) if no point of P\A lies in the convex hull of A. We show that for any k, n there is an N = N (k, n) such that any k-convex set of at least N points in general position in the plane contains an empty n-gon. We also prove an analogous statement in R d for each odd d ≥ 3.