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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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