Crossing families

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

doi:10.1007/BF01215345

Crossing families

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

Research output: Contribution to journal › Article › peer-review

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 language	English (US)
Pages (from-to)	127-134
Number of pages	8
Journal	Combinatorica
Volume	14
Issue number	2
DOIs	https://doi.org/10.1007/BF01215345
State	Published - Jun 1994

Keywords

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

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Computational Mathematics

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10.1007/BF01215345

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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.",

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T1 - Crossing families

AU - Aronov, B.

AU - Erdos, P.

AU - Goddard, W.

AU - Kleitman, D. J.

AU - Klugerman, M.

AU - Pach, J.

AU - Schulman, L. J.

PY - 1994/6

Y1 - 1994/6

N2 - 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.

AB - 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.

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Crossing families

Abstract

Keywords

ASJC Scopus subject areas

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