The Erdos-Szekeres Conjecture Revisited

The famous and still open Erdos-Szekeres Conjecture from 1935 states that every set of at least 2k-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 Erdos-Szekeres Conjecture by showing that every set of at least 2k-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 2k-2 + 1 from the Erdos-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 Erdos-Szekeres Conjecture is not true. Finally, we prove the Erdos-Szekeres Conjecture for so-called decomposable sets and provide new constructions of sets of 2k-2 points without k points in convex position, generalizing all previously known constructions of such point sets and allowing us to computationally tackle the Erdos-Szekeres Conjecture for large values of k.