A superlinear lower bound on the number of 5-holes

Result code in 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>
Result on the web
<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>

Result language
angličtina
Original language name
A superlinear lower bound on the number of 5-holes
Original language description
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.
Czech name
—
Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
Result was created during the realization of more than one project. More information in the Projects tab.
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

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