The partitioned version of the Erdős-Szekeres theorem (Q1864120)

For \(k\geq 4\), a finite planar point set \(X\) is called a convex \(k\)-clustering if it is a disjoint union of \(k\) sets \(X_1, \ldots , X_k\) of equal sizes such that \(x_1x_2\ldots x_k\) is a convex \(k\)-gon for each choice of \(x_1\in X_1,\ldots , x_k\in X_k\). The authors answer a question of Gil Kalai: they show that for every \(k\geq 4\) 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. Their result strengthens the so-called positive fraction Erdős-Szekeres theorem proved by Bárány and Valtr. The proof gives reasonable estimates on \(c\) and \(c'\), and it works also in higher dimensions. The authors also improve the previous constants for the positive fraction Erdős-Szekeres theorem obtained by Pach and Solymosi.