### Abstract

A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any two of them intersect and no three curves pass through the same point, is at least (1-O (1))n^{2}. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have precisely one point in common, at which they do not properly cross.) An important ingredient of our proofs is the following statement: Let S be a family of the graphs of n continuous real functions defined on ℝ, no three of which pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is ω (nt √logt/log log t).

Original language | English |
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Title of host publication | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |

Publisher | Association for Computing Machinery |

Pages | 1506-1516 |

Number of pages | 11 |

Volume | 2015-January |

Edition | January |

Publication status | Published - 2015 |

Event | 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 - San Diego, United States Duration: Jan 4 2015 → Jan 6 2015 |

### Other

Other | 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 |
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Country | United States |

City | San Diego |

Period | 1/4/15 → 1/6/15 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms*(January ed., Vol. 2015-January, pp. 1506-1516). Association for Computing Machinery.

**On the Richter-Thomassen conjecture about pairwise intersecting closed curves.** / Pach, Janos; Rubin, Natan; Tardos, G.

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

*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms.*January edn, vol. 2015-January, Association for Computing Machinery, pp. 1506-1516, 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, United States, 1/4/15.

}

TY - GEN

T1 - On the Richter-Thomassen conjecture about pairwise intersecting closed curves

AU - Pach, Janos

AU - Rubin, Natan

AU - Tardos, G.

PY - 2015

Y1 - 2015

N2 - A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any two of them intersect and no three curves pass through the same point, is at least (1-O (1))n2. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have precisely one point in common, at which they do not properly cross.) An important ingredient of our proofs is the following statement: Let S be a family of the graphs of n continuous real functions defined on ℝ, no three of which pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is ω (nt √logt/log log t).

AB - A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any two of them intersect and no three curves pass through the same point, is at least (1-O (1))n2. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have precisely one point in common, at which they do not properly cross.) An important ingredient of our proofs is the following statement: Let S be a family of the graphs of n continuous real functions defined on ℝ, no three of which pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is ω (nt √logt/log log t).

UR - http://www.scopus.com/inward/record.url?scp=84938242774&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84938242774&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84938242774

VL - 2015-January

SP - 1506

EP - 1516

BT - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

PB - Association for Computing Machinery

ER -