Discrete & Computational Geometry

, Volume 28, Issue 4, pp 625–637 | Cite as

The Partitioned Version of the Erdős—Szekeres Theorem

  • Pór
  • Valtr
Article

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 X1, . . . ,Xk of equal sizes such that x1x2 . . . xk is a convex k -gon for each choice of x1∈ X1, . . . ,xk∈ Xk . 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.

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Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  • Pór
    • 1
  • Valtr
    • 2
  1. 1.Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, 1364 Budapest, Hungary apor@renyi.hu Hungary
  2. 2.Department of Applied Mathematics, and Institute for Theoretical Computer Science (ITI), Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic valtr@kam.mff.cuni.czCzech Republic

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