On Erdős-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%2F19%3A10403065" target="_blank" >RIV/00216208:11320/19:10403065 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/978-3-030-25005-8_4" target="_blank" >https://doi.org/10.1007/978-3-030-25005-8_4</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/978-3-030-25005-8_4" target="_blank" >10.1007/978-3-030-25005-8_4</a>

Result language
angličtina
Original language name
On Erdős-Szekeres-type problems for k-convex point sets
Original language description
We study Erdős-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 Erdős-Szekeres Theorem by showing that, for every fixed k 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 Ω(logk 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(log n). This gives a solution to a problem posed by Aichholzer et al. We prove that there is a constant c > 0 such that, for every n in 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 . log n points from S contains a point of S in its interior. This matches an earlier upper bound by Aichholzer et al. up to a multiplicative constant and answers another of their open problems.
Czech name
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Czech description
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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
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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