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Abstract: The famous and still open Erdős–Szekeres Conjecture from 1935 states that every set of at least 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 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 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 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.

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