The Erdos-Szekeres Conjecture Revisited

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F25%3A10510331" target="_blank" >RIV/00216208:11320/25:10510331 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.4230/LIPIcs.SoCG.2025.13" target="_blank" >http://dx.doi.org/10.4230/LIPIcs.SoCG.2025.13</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.SoCG.2025.13" target="_blank" >10.4230/LIPIcs.SoCG.2025.13</a>

Jazyk výsledku

angličtina

Název v původním jazyce

The Erdos-Szekeres Conjecture Revisited

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

The Erdos-Szekeres Conjecture Revisited

Popis výsledku anglicky

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.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GX23-04949X" target="_blank" >GX23-04949X: Stěžejní otázky diskrétní geometrie</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

41st International Symposium on Computational Geometry (SoCG 2025)

ISBN

978-3-95977-370-6

ISSN

—

e-ISSN

—

Počet stran výsledku

15

Strana od-do

—

Název nakladatele

Leibniz International Proceedings in Informatics, LIPIcs

Místo vydání

Neuveden

Místo konání akce

Kanazawa, Japonsko

Datum konání akce

23. 6. 2025

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

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