## Abstract

If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a touching point. The main result of this paper is a Crossing Lemma for simple curves: Let X and T stand for the sets of intersection points and touching points, respectively, in a family of n simple curves in the plane, no three of which pass through the same point. If |T|>cn, for some fixed constant c>0, then we prove that |X|=Ω(|T|(loglog(|T|/n))^{1/504}). In particular, if |T|/n→∞ then the number of intersection points is much larger than the number of touching points. As a corollary, we confirm the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between n pairwise intersecting simple closed (i.e., Jordan) curves in the plane, no three of which pass through the same point, is at least (1−o(1))n^{2}.

Original language | English |
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Pages (from-to) | 908-940 |

Number of pages | 33 |

Journal | Advances in Mathematics |

Volume | 331 |

DOIs | |

State | Published - 20 Jun 2018 |

## Keywords

- Arrangements of curves
- Combinatorial geometry
- Contact graphs
- Crossing Lemma
- Extremal problems
- Separators

## ASJC Scopus subject areas

- General Mathematics