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The Erdős-Szekeres Conjecture Revisited

Authors Jineon Baek , Martin Balko

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Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Computational geometry
  • Mathematics of computing → Combinatorics
Keywords
  • convex position
  • Erdős-Szekeres theorem
  • point set

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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.

Cite As Get BibTex

Jineon Baek and Martin Balko. The Erdős-Szekeres Conjecture Revisited. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.13

Author Details

Jineon Baek
  • Department of Mathematics, University of Michigan, Ann Arbor, MI, USA
Martin Balko
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

Funding

  • Baek, Jineon: acknowledges support from the Korea Foundation for Advanced Studies during the completion of this research.
  • Balko, Martin: supported by grant no. 23-04949X of the Czech Science Foundation (GAČR) and by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE 24/SCI/008).

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