The Erdős-Szekeres Conjecture Revisited

Jineon Baek, Martin Balko

Abstract: The famous and still open Erdős-Szekeres Conjecture from 1935 states that every set of at least 2^{k-2}+1 points in the plane with no three being collinear contains k points in convex position, that is, k points that are vertices of a convex polygon. In this paper, we revisit this conjecture and show several new related results. First, we prove a relaxed version of the Erdős-Szekeres Conjecture by showing that every set of at least 2^{k-2}+1 points in the plane with no three being collinear contains a split k-gon, a relaxation of k-tuple of points in convex position. Moreover, we show that this is tight, showing that the value 2^{k-2}+1 from the Erdős-Szekeres Conjecture is exactly the right threshold for split k-gons. We obtain an analogous relaxation in a much more general setting of ordered 3-uniform hypergraphs where we also show that, perhaps surprisingly, a corresponding generalization of the Erdős-Szekeres Conjecture is not true. Finally, we prove the Erdős-Szekeres Conjecture for so-called decomposable sets and provide new constructions of sets of 2^{k-2} points without k points in convex position, generalizing all previously known constructions of such point sets and allowing us to computationally tackle the Erdős-Szekeres Conjecture for large values of k.