On the positive fraction Erdős–Szekeres theorem for convex sets

Let F = { F 1 , … , F n } be a collection of disjoint compact convex sets in the plane. We say that F is in general position if no F i is in the convex hull of two other F i ’s. We say that F is in convex position if no F i is in the convex hull of the other n − 1 F i ’s. For k ≥ 4 , F is called a k -cluster if it is a disjoint union of k subfamilies F 1 , F 2 , … , F k ⊂ F of equal size such that each transversal { F 1 , F 2 , … , F k } , F i ∈ F i , 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 − 37.8 k − 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].