Combinatorica

, Volume 14, Issue 2, pp 127–134 | Cite as

Crossing families

  • B. Aronov
  • P. Erdős
  • W. Goddard
  • D. J. Kleitman
  • M. Klugerman
  • J. Pach
  • L. J. Schulman
Article

Abstract

Given a set of points in the plane, a crossing family is a collection of line segments, each joining two of the points, such that any two line segments intersect internally. Two setsA andB of points in the plane are mutually avoiding if no line subtended by a pair of points inA intersects the convex hull ofB, and vice versa. We show that any set ofn points in general position contains a pair of mutually avoiding subsets each of size at least\(\sqrt {n/12} \). As a consequence we show that such a set possesses a crossing family of size at least\(\sqrt {n/12} \), and describe a fast algorithm for finding such a family.

AMS subject classification code (1991)

52 C 10 68 Q 20 

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Copyright information

© Akadémiai Kiadó 1994

Authors and Affiliations

  • B. Aronov
  • P. Erdős
    • 1
  • W. Goddard
    • 2
  • D. J. Kleitman
    • 2
  • M. Klugerman
    • 2
  • J. Pach
    • 1
    • 3
  • L. J. Schulman
    • 2
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesHungary
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyUSA
  3. 3.Courant InstituteNew York UniversityNew YorkUSA

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