### 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 |
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Title of host publication | STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Moses Charikar, Edith Cohen |

Publisher | Association for Computing Machinery |

Pages | 1158-1166 |

Number of pages | 9 |

ISBN (Electronic) | 9781450367059 |

DOIs | |

Publication status | Published - Jun 23 2019 |

Event | 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 - Phoenix, United States Duration: Jun 23 2019 → Jun 26 2019 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 |
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Country | United States |

City | Phoenix |

Period | 6/23/19 → 6/26/19 |

### Keywords

- Avoiding edges
- Comparability graphs
- Computational geometry
- Crossing edges
- Extremal combinatorics
- Geometric graphs
- Intersection graphs
- Partial orders

### ASJC Scopus subject areas

- Software

### Cite this

*STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing*(pp. 1158-1166). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3313276.3316328

**Planar point sets determine many pairwise crossing segments.** / Pach, János; Rubin, Natan; Tardos, G.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing.*Proceedings of the Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, pp. 1158-1166, 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, United States, 6/23/19. https://doi.org/10.1145/3313276.3316328

}

TY - GEN

T1 - Planar point sets determine many pairwise crossing segments

AU - Pach, János

AU - Rubin, Natan

AU - Tardos, G.

PY - 2019/6/23

Y1 - 2019/6/23

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 edges

KW - Comparability graphs

KW - Computational geometry

KW - Crossing edges

KW - Extremal combinatorics

KW - Geometric graphs

KW - Intersection graphs

KW - Partial orders

UR - http://www.scopus.com/inward/record.url?scp=85068033615&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068033615&partnerID=8YFLogxK

U2 - 10.1145/3313276.3316328

DO - 10.1145/3313276.3316328

M3 - Conference contribution

AN - SCOPUS:85068033615

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 1158

EP - 1166

BT - STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

A2 - Charikar, Moses

A2 - Cohen, Edith

PB - Association for Computing Machinery

ER -