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Summary: Piercing convex sets and the Hadwiger Debrunner (p, q)-problem
Noga Alon
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
and
Daniel J. Kleitman
Department of Mathematics
MIT, Cambridge, Ma, 02139
Abstract
A family of sets has the (p, q) property if among any p members of the family some q have
a nonempty intersection. It is shown 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. This settles an old problem
of Hadwiger and Debrunner.
Research supported in part by a United States Israel BSF Grant and by a Bergmann Memorial Grant
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