A purely combinatorial proof of the Hadwiger Debrunner (p, q) conjecture

N. Alon; D. J. Kleitman

A purely combinatorial proof of the Hadwiger Debrunner (p, q) conjecture

N. Alon, D. J. Kleitman

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

Abstract

A family of sets has the (p, q) property if among any p members of the family some q have a nonempty intersection. The authors have proved that for every p ≥ q ≥ d + 1 there is a c = c(p, q, d) < ∞ such that for every family F of compact, convex sets in R^d which has the (p, q) property there is a set of at most c points in R^d that intersects each member of F, thus settling an old problem of Hadwiger and Debrunner. Here we present a purely combinatorial proof of this result.

Original language	English (US)
Article number	R1
Pages (from-to)	1-8
Number of pages	8
Journal	Electronic Journal of Combinatorics
Volume	4
Issue number	2 R
State	Published - 1997
Externally published	Yes

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Applied Mathematics

Cite this

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AB - A family of sets has the (p, q) property if among any p members of the family some q have a nonempty intersection. The authors have proved that for every p ≥ q ≥ d + 1 there is a c = c(p, q, d) < ∞ such that for every family F of compact, convex sets in Rd which has the (p, q) property there is a set of at most c points in Rd that intersects each member of F, thus settling an old problem of Hadwiger and Debrunner. Here we present a purely combinatorial proof of this result.

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A purely combinatorial proof of the Hadwiger Debrunner (p, q) conjecture

Abstract

All Science Journal Classification (ASJC) codes

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