On the positive fraction Erdős–Szekeres theorem for convex sets

Let F={F1,…,Fn}F={F1,…,Fn} be a collection of disjoint compact convex sets in the plane. We say that FF is in general position   if no FiFi is in the convex hull of two other FiFi’s. We say that FF is in convex position   if no FiFi is in the convex hull of the other n−1n−1FiFi’s. For k≥4k≥4, FF is called a kk-cluster   if it is a disjoint union of kk subfamilies F1,F2,…,Fk⊂FF1,F2,…,Fk⊂F of equal size such that each transversal {F1,F2,…,Fk}{F1,F2,…,Fk}, Fi∈FiFi∈Fi, is in convex position. In this paper we show that for any FF in general position there is a kk-cluster F′⊂FF′⊂F of size at least 2−37.8k−o(1)|F|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].