The Partitioned Version of the Erdős—Szekeres Theorem

Abstract

Let k≥ 4. A finite planar point set X is called a convex k -clustering if it is a disjoint union of k sets X 1 , . . . ,X k of equal sizes such that x 1 x 2 . . . x k is a convex k -gon for each choice of x 1 ∈ X 1 , . . . ,x k ∈ X k . Answering a question of Gil Kalai, we show that for every k≥ 4 there are two constants c=c(k) , c’=c’(k) such that the following holds. If X is a finite set of points in general position in the plane, then it has a subset X’ of size at most c’ such that X \ X’ can be partitioned into at most c convex k -clusterings. The special case k=4 was proved earlier by Pór. Our result strengthens the so-called positive fraction Erdos—Szekeres theorem proved by Barany and Valtr. The proof gives reasonable estimates on c and c’ , and it works also in higher dimensions. We also improve the previous constants for the positive fraction Erdos—Szekeres theorem obtained by Pach and Solymosi.

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Pór, Valtr The Partitioned Version of the Erdős—Szekeres Theorem. Discrete Comput Geom 28, 625–637 (2002). https://doi.org/10.1007/s00454-002-2894-1

Download citation