Crossing families

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

    Research output: Contribution to journalArticle

    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 (US)
    Pages (from-to)127-134
    Number of pages8
    JournalCombinatorica
    Volume14
    Issue number2
    DOIs
    StatePublished - Jun 1994

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    Joining
    Line segment
    Intersect
    Convex Hull
    Set of points
    Fast Algorithm
    Subset
    Family
    Line

    Keywords

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

    ASJC Scopus subject areas

    • Discrete Mathematics and Combinatorics
    • Mathematics(all)

    Cite this

    Aronov, B., Erdos, 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

    Crossing families. / Aronov, B.; Erdos, P.; Goddard, W.; Kleitman, D. J.; Klugerman, M.; Pach, J.; Schulman, L. J.

    In: Combinatorica, Vol. 14, No. 2, 06.1994, p. 127-134.

    Research output: Contribution to journalArticle

    Aronov, B, Erdos, P, Goddard, W, Kleitman, DJ, Klugerman, M, Pach, J & Schulman, LJ 1994, 'Crossing families', Combinatorica, vol. 14, no. 2, pp. 127-134. https://doi.org/10.1007/BF01215345
    Aronov B, Erdos P, Goddard W, Kleitman DJ, Klugerman M, Pach J et al. Crossing families. Combinatorica. 1994 Jun;14(2):127-134. https://doi.org/10.1007/BF01215345
    Aronov, B. ; Erdos, P. ; Goddard, W. ; Kleitman, D. J. ; Klugerman, M. ; Pach, J. ; Schulman, L. J. / Crossing families. In: Combinatorica. 1994 ; Vol. 14, No. 2. pp. 127-134.
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