Planar Point Sets Determine Many Pairwise Crossing Segments

János Pach; Natan Rubin; Gábor Tardos

Planar Point Sets Determine Many Pairwise Crossing Segments

Combinatorics 2023-05-02 v2 Computational Geometry

Authors: János Pach , Natan Rubin , Gábor Tardos

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, $\Omega\left(\sqrt n\right)$ , was proved more than 25 years ago by Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.

Keywords

incidence geometry of points and lines

Cite

@article{arxiv.1904.08845,
  title  = {Planar Point Sets Determine Many Pairwise Crossing Segments},
  author = {János Pach and Natan Rubin and Gábor Tardos},
  journal= {arXiv preprint arXiv:1904.08845},
  year   = {2023}
}

Comments

A preliminary version to appear in the proceedings of STOC 2019

Planar Point Sets Determine Many Pairwise Crossing Segments

Abstract

Keywords

Cite

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