2026-07-21T03:13:48Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23165 2025-10-02T12:41:40Z ddc:004 open_access

The Erdős-Szekeres Conjecture Revisited Baek, Jineon Balko, Martin convex position Erdős-Szekeres theorem point set 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jineon Baek and Martin Balko 2025 Is Part Of LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2025.13 urn:nbn:de:0030-drops-231655 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.13 eng https://creativecommons.org/licenses/by/4.0/legalcode