The Partitioned Version of the Erd?s—Szekeres Theorem |
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Authors: | Pór Valtr |
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Institution: | (1) Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, 1364 Budapest, Hungary apor@renyi.hu , HU;(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.cz, CZ |
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Abstract: |
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. |
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Keywords: | |
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