String graphs have the Erdős–Hajnal property

Abstract

The following Ramsey-type question is one of the central problems in combinatorial geometry. Given a collection of certain geometric objects in the plane (e.g. segments, rectangles, convex sets, arcwise connected sets) of size n , what is the size of the largest subcollection in which either any two elements have a nonempty intersection, or any two elements are disjoint? We prove that there exists an absolute constant c>0 such that if \mathcal{C} is a collection of n curves in the plane, then \mathcal{C} contains at least n^{c} elements that are pairwise intersecting, or n^{c} elements that are pairwise disjoint. This resolves a problem raised by Alon, Pach, Pinchasi, Radoičić and Sharir, and Fox and Pach. Furthermore, as any geometric object can be arbitrarily closely approximated by a curve, this shows that the answer to the aforementioned question is at least n^{c} for any collection of n geometric objects.