On crossing-families in planar point sets

Oswin Aichholzer; Jan Kynčl; Manfred Scheucher; Birgit Vogtenhuber; Pavel Valtr

doi:10.1016/j.comgeo.2022.101899

On crossing-families in planar point sets

Oswin Aichholzer^*, Jan Kynčl, Manfred Scheucher, Birgit Vogtenhuber, Pavel Valtr

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

A k-crossing family in a point set S in general position is a set of k segments spanned by points of S such that all k segments mutually cross. In this short note we present two statements on crossing families which are based on sets of small cardinality: (1) Any set of at least 15 points contains a crossing family of size 4. (2) There are sets of n points which do not contain a crossing family of size larger than [Formula presented]. Both results improve the previously best known bounds.

Original language	English
Article number	101899
Journal	Computational Geometry: Theory and Applications
Volume	107
DOIs	https://doi.org/10.1016/j.comgeo.2022.101899
Publication status	Published - Dec 2022

Keywords

Crossing family
Geometric thrackle
Order type
Point set

ASJC Scopus subject areas

Computer Science Applications
Geometry and Topology
Control and Optimization
Computational Theory and Mathematics
Computational Mathematics

Fields of Expertise

Information, Communication & Computing

Access to Document

10.1016/j.comgeo.2022.101899

Cite this

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title = "On crossing-families in planar point sets",

abstract = "A k-crossing family in a point set S in general position is a set of k segments spanned by points of S such that all k segments mutually cross. In this short note we present two statements on crossing families which are based on sets of small cardinality: (1) Any set of at least 15 points contains a crossing family of size 4. (2) There are sets of n points which do not contain a crossing family of size larger than [Formula presented]. Both results improve the previously best known bounds.",

keywords = "Crossing family, Geometric thrackle, Order type, Point set",

author = "Oswin Aichholzer and Jan Kyn{\v c}l and Manfred Scheucher and Birgit Vogtenhuber and Pavel Valtr",

note = "Publisher Copyright: {\textcopyright} 2022 Elsevier B.V.",

year = "2022",

month = dec,

doi = "10.1016/j.comgeo.2022.101899",

language = "English",

volume = "107",

journal = "Computational Geometry: Theory and Applications",

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TY - JOUR

T1 - On crossing-families in planar point sets

AU - Aichholzer, Oswin

AU - Kynčl, Jan

AU - Scheucher, Manfred

AU - Vogtenhuber, Birgit

AU - Valtr, Pavel

PY - 2022/12

Y1 - 2022/12

N2 - A k-crossing family in a point set S in general position is a set of k segments spanned by points of S such that all k segments mutually cross. In this short note we present two statements on crossing families which are based on sets of small cardinality: (1) Any set of at least 15 points contains a crossing family of size 4. (2) There are sets of n points which do not contain a crossing family of size larger than [Formula presented]. Both results improve the previously best known bounds.

AB - A k-crossing family in a point set S in general position is a set of k segments spanned by points of S such that all k segments mutually cross. In this short note we present two statements on crossing families which are based on sets of small cardinality: (1) Any set of at least 15 points contains a crossing family of size 4. (2) There are sets of n points which do not contain a crossing family of size larger than [Formula presented]. Both results improve the previously best known bounds.

KW - Crossing family

KW - Geometric thrackle

KW - Order type

KW - Point set

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U2 - 10.1016/j.comgeo.2022.101899

DO - 10.1016/j.comgeo.2022.101899

M3 - Article

AN - SCOPUS:85133963879

SN - 0925-7721

VL - 107

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

M1 - 101899

ER -

On crossing-families in planar point sets

Abstract

Keywords

ASJC Scopus subject areas

Fields of Expertise

Access to Document

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