On Erdos-Szekeres-type problems for k-convex point sets

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10420185" target="_blank" >RIV/00216208:11320/20:10420185 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=3D83ELsX7b" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=3D83ELsX7b</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.ejc.2020.103157" target="_blank" >10.1016/j.ejc.2020.103157</a>

Jazyk výsledku

angličtina

Název v původním jazyce

On Erdos-Szekeres-type problems for k-convex point sets

Popis výsledku v původním jazyce

We study Erdos-Szekeres-type problems for k-convex point sets, a recently introduced notion that naturally extends the concept of convex position. A finite set S of n points is k-convex if there exists a spanning simple polygonization of S such that the intersection of any straight line with its interior consists of at most k connected components. We address several open problems about k-convex point sets. In particular, we extend the well-known Erdos-Szekeres Theorem by showing that, for every fixed k is an element of N, every set of n points in the plane in general position (with no three collinear points) contains a k-convex subset of size at least Omega(log(k) n). We also show that there are arbitrarily large 3-convex sets of n points in the plane in general position whose largest 1-convex subset has size O(logn). This gives a solution to a problem posed by Aichholzer et al. (2014). We prove that there is a constant c &gt; 0 such that, for every n is an element of N, there is a set S of n points in the plane in general position such that every 2-convex polygon spanned by at least c . logn points from S contains a point of S in its interior. This matches an earlier upper bound by Aichholzer et al. (2014) up to a multiplicative constant and answers another of their open problems. (C) 2020 Elsevier Ltd. All rights reserved.

Název v anglickém jazyce

On Erdos-Szekeres-type problems for k-convex point sets

Popis výsledku anglicky

We study Erdos-Szekeres-type problems for k-convex point sets, a recently introduced notion that naturally extends the concept of convex position. A finite set S of n points is k-convex if there exists a spanning simple polygonization of S such that the intersection of any straight line with its interior consists of at most k connected components. We address several open problems about k-convex point sets. In particular, we extend the well-known Erdos-Szekeres Theorem by showing that, for every fixed k is an element of N, every set of n points in the plane in general position (with no three collinear points) contains a k-convex subset of size at least Omega(log(k) n). We also show that there are arbitrarily large 3-convex sets of n points in the plane in general position whose largest 1-convex subset has size O(logn). This gives a solution to a problem posed by Aichholzer et al. (2014). We prove that there is a constant c &gt; 0 such that, for every n is an element of N, there is a set S of n points in the plane in general position such that every 2-convex polygon spanned by at least c . logn points from S contains a point of S in its interior. This matches an earlier upper bound by Aichholzer et al. (2014) up to a multiplicative constant and answers another of their open problems. (C) 2020 Elsevier Ltd. All rights reserved.

Druh

J_imp - Článek v periodiku v databázi Web of Science

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/GA18-19158S" target="_blank" >GA18-19158S: Algoritmické, strukturální a složitostní aspekty geometrických a dalších konfigurací</a><br>

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

European Journal of Combinatorics

ISSN

0195-6698

e-ISSN

—

Svazek periodika

89

Číslo periodika v rámci svazku

27 May

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

22

Strana od-do

103157

Kód UT WoS článku

000556551000015

EID výsledku v databázi Scopus

2-s2.0-85085320484