On the positive fraction Erdös-Szekeres theorem for convex sets

Abstract

Let F = {F1, ..., Fn} be a collection of disjoint compact convex sets in the plane. We say that F is in general position if no Fi is in the convex hull of two other Fi's. We say that F is in convex position if no Fi is in the convex hull of the other n - 1Fi's. For k ≥ 4, F is called a k-cluster if it is a disjoint union of k subfamilies F1, F2, ..., Fk ⊂ F of equal size such that each transversal {F1, F2,...,Fk}, Fi ∈ Fi, is in convex position. In this paper we show that for any F in general position there is a k-cluster F' ⊂ F of size at least 2-37.8k-o(1)|F|. This improves the result of J. Pach and J. Solymosi [Canonical theorems for convex sets, Discrete and Computational Geometry 19 (1998) 427-435].