Planar point sets determine many pairwise crossing segments

János Pach, Natan Rubin, Gábor Tardos

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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.

Original languageEnglish
Article number107779
JournalAdvances in Mathematics
StatePublished - 6 Aug 2021


  • Avoiding sets
  • Crossing families
  • Epsilon-nets
  • Geometric graphs
  • Intersection graphs
  • Partial orders

ASJC Scopus subject areas

  • Mathematics (all)


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