# The Partitioned Version of the Erdős—Szekeres Theorem

- First Online:

DOI: 10.1007/s00454-002-2894-1

- Cite this article as:
- Pór & Valtr Discrete Comput Geom (2002) 28: 625. doi:10.1007/s00454-002-2894-1

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## 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.