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

Department of Computer Science

Ben-Gurion University of the Negev

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

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	American 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

All Science Journal Classification (ASJC) codes

General Mathematics

Access to Document

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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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.

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KW - Crossing families

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KW - Geometric graphs

KW - Intersection graphs

KW - Partial orders

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Planar point sets determine many pairwise crossing segments

Abstract

Keywords

All Science Journal Classification (ASJC) codes

Access to Document

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