A superlinear lower bound on the number of 5-holes

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10420187" target="_blank" >RIV/00216208:11320/20:10420187 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=1RqV4UMqjy" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=1RqV4UMqjy</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jcta.2020.105236" target="_blank" >10.1016/j.jcta.2020.105236</a>

Jazyk výsledku

angličtina

Název v původním jazyce

A superlinear lower bound on the number of 5-holes

Popis výsledku v původním jazyce

Let P be a finite set of points in the plane in general position, that is, no three points of Pare on a common line. We say that a set H of five points from Pis a 5-hole in P if His the vertex set of a convex 5-gon containing no other points of P. For a positive integern, let h(5)(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h(5)(n) have been of order Omega(n) and O(n(2)), respectively. We show that h(5)(n) = Omega(n log(4/5)n), obtaining the first superlinear lower bound on h(5)(n). The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set Pof points in the plane in general position is partitioned by a line l into two subsets, each of size at least 5 and not in convex position, then l intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted. (c) 2020 Elsevier Inc. All rights reserved.

Název v anglickém jazyce

A superlinear lower bound on the number of 5-holes

Popis výsledku anglicky

Let P be a finite set of points in the plane in general position, that is, no three points of Pare on a common line. We say that a set H of five points from Pis a 5-hole in P if His the vertex set of a convex 5-gon containing no other points of P. For a positive integern, let h(5)(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h(5)(n) have been of order Omega(n) and O(n(2)), respectively. We show that h(5)(n) = Omega(n log(4/5)n), obtaining the first superlinear lower bound on h(5)(n). The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set Pof points in the plane in general position is partitioned by a line l into two subsets, each of size at least 5 and not in convex position, then l intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted. (c) 2020 Elsevier Inc. All rights reserved.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

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

Rok uplatnění

2020

Kód důvěrnosti údajů

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

Název periodika

Journal of Combinatorial Theory - Series A

ISSN

0097-3165

e-ISSN

—

Svazek periodika

173

Číslo periodika v rámci svazku

February 27

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

31

Strana od-do

105236

Kód UT WoS článku

000527891300014

EID výsledku v databázi Scopus

2-s2.0-85079878658