### 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 |
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Pages (from-to) | 127-134 |

Number of pages | 8 |

Journal | Combinatorica |

Volume | 14 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1994 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)

### Cite this

*Combinatorica*,

*14*(2), 127-134. https://doi.org/10.1007/BF01215345

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

Research output: Contribution to journal › Article

*Combinatorica*, vol. 14, no. 2, pp. 127-134. https://doi.org/10.1007/BF01215345

}

TY - JOUR

T1 - Crossing families

AU - Aronov, B.

AU - Erdős, 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.

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

UR - http://www.scopus.com/inward/record.url?scp=0003501716&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0003501716&partnerID=8YFLogxK

U2 - 10.1007/BF01215345

DO - 10.1007/BF01215345

M3 - Article

VL - 14

SP - 127

EP - 134

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 2

ER -