Planar point sets determine many pairwise crossing segments

János Pach; Natan Rubin; Gábor Tardos

doi:10.1016/j.aim.2021.107779

Planar point sets determine many pairwise crossing segments

János Pach
, Natan Rubin
, Gábor Tardos

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

17 Scopus citations

Abstract

We show that any set of n points in general position in the plane determines n^1−o(1) pairwise crossing segments. The best previously known lower bound, Ω(n), was proved more than 25 years ago by Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.

Original language	English
Article number	107779
Journal	Advances in Mathematics
Volume	386
DOIs	https://doi.org/10.1016/j.aim.2021.107779
State	Published - 6 Aug 2021

Keywords

Avoiding sets
Crossing families
Epsilon-nets
Geometric graphs
Intersection graphs
Partial orders

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2021.107779

Cite this

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title = "Planar point sets determine many pairwise crossing segments",

abstract = "We show that any set of n points in general position in the plane determines n1−o(1) pairwise crossing segments. The best previously known lower bound, Ω(n), was proved more than 25 years ago by Aronov, Erd{\H o}s, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.",

keywords = "Avoiding sets, Crossing families, Epsilon-nets, Geometric graphs, Intersection graphs, Partial orders",

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AU - Rubin, Natan

AU - Tardos, Gábor

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N2 - We show that any set of n points in general position in the plane determines n1−o(1) pairwise crossing segments. The best previously known lower bound, Ω(n), was proved more than 25 years ago by Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.

AB - We show that any set of n points in general position in the plane determines n1−o(1) pairwise crossing segments. The best previously known lower bound, Ω(n), was proved more than 25 years ago by Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.

KW - Avoiding sets

KW - Crossing families

KW - Epsilon-nets

KW - Geometric graphs

KW - Intersection graphs

KW - Partial orders

UR - https://www.scopus.com/pages/publications/85107650177

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DO - 10.1016/j.aim.2021.107779

M3 - Article

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SN - 0001-8708

VL - 386

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107779

ER -

Planar point sets determine many pairwise crossing segments

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

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