Crossing families

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

Research output: Contribution to journalArticle

38 Citations (Scopus)

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 sets A and B of points in the plane are mutually avoiding if no line subtended by a pair of points in A intersects the convex hull of B, and vice versa. We show that any set of n points in general position contains a pair of mutually avoiding subsets each of size at least {Mathematical expression}. As a consequence we show that such a set possesses a crossing family of size at least {Mathematical expression}, and describe a fast algorithm for finding such a family.

Original languageEnglish
Pages (from-to)127-134
Number of pages8
JournalCombinatorica
Volume14
Issue number2
DOIs
Publication statusPublished - Jun 1994

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Keywords

  • AMS subject classification code (1991): 52C10, 68Q20

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

Cite this

Aronov, B., Erdős, P., Goddard, W., Kleitman, D. J., Klugerman, M., Pach, J., & Schulman, L. J. (1994). Crossing families. Combinatorica, 14(2), 127-134. https://doi.org/10.1007/BF01215345