A Crossing Lemma for Jordan curves

János Pach, Natan Rubin, Gábor Tardos

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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|(log⁡log⁡(|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))n2.

Original languageEnglish
Pages (from-to)908-940
Number of pages33
JournalAdvances in Mathematics
StatePublished - 20 Jun 2018


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

ASJC Scopus subject areas

  • General Mathematics


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