TY - JOUR

T1 - A purely combinatorial proof of the Hadwiger Debrunner (p, q) conjecture

AU - Alon, N.

AU - Kleitman, D. J.

PY - 1997

Y1 - 1997

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=0007372644&partnerID=8YFLogxK

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AN - SCOPUS:0007372644

SN - 1077-8926

VL - 4

SP - 1

EP - 8

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 2 R

M1 - R1

ER -