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

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

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

Original language description

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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

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

Project

<a href="/en/project/GA18-19158S" target="_blank" >GA18-19158S: Algorithmic, structural and complexity aspects of geometric and other configurations</a><br>

Continuities

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

Publication year

2020

Confidentiality

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

Name of the periodical

European Journal of Combinatorics

ISSN

0195-6698

e-ISSN

—

Volume of the periodical

89

Issue of the periodical within the volume

27 May

Country of publishing house

GB - UNITED KINGDOM

Number of pages

22

Pages from-to

103157

UT code for WoS article

000556551000015

EID of the result in the Scopus database

2-s2.0-85085320484