Erdős–Szekeres Theorems for Families of Convex Sets

Abstract

The well-known Erdos-Szekeres theorem states that every sufficiently large set of points in the plane containing no three points on a line, has a large subset in convex position. This classical result has been generalized in several directions. In this article we review recent progress related to one such direction, initiated by Bisztriczky and Fejes Toth, in which the points are replaced by convex sets.