Crossing families


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.

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  1. [1]

    N. Alon andP. Erdős: Disjoint edges in geometric graphs,Discrete Comput. Geom. 4 (1989), 287–290.

    Google Scholar 

  2. [2]

    V. Capoyleas andJ. Pach: A Turán-type theorem on chords of a convex polygon,J. Combin. Theory. Ser. B 56 (1992), 9–15.

    Google Scholar 

  3. [3]

    H. Edelsbrunner:Algorithms in Combinatorial Geometry, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  4. [4]

    P. Erdős andG. Szekeres: A combinatorial problem in geometry,Compositio Math. 2 (1935), 463–470.

    Google Scholar 

  5. [5]

    J. Matoušek: Efficient partition tress,Discrete Comput. Geom. 8 (1992), 315–334.

    Google Scholar 

  6. [6]

    J. Pach andJ. Törőcsik: Some geometric applications of Dilworth's theorem,Proc. 9th ACM Symp. Comput. Geom. (1993), 264–269. Also:Discrete Comput. Geom., to appear.

  7. [7]

    P. Valtr: On mutually avoiding sets, manuscript.

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Research supported in part by DARPA grant N00014-89-J-1988, Air Force AFOSR-89-0271, NSF grant DMS-8606225, and an ONR graduate fellowship. Further, part of this work was conducted at and supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center-NSF-STC8809648.

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Aronov, B., Erdős, P., Goddard, W. et al. Crossing families. Combinatorica 14, 127–134 (1994).

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AMS subject classification code (1991)

  • 52 C 10
  • 68 Q 20