 
Summary: Bounding the piercing number
Noga Alon
Gil Kalai
Abstract
It is shown that for every k and every p q d + 1 there is a c = c(k, p, q, d) < such
that the following holds. For every family H whose members are unions of at most k compact,
convex sets in Rd
in which any set of p members of the family contains a subset of cardinality
q with a nonempty intersection there is a set of at most c points in Rd
that intersects each
member of H. It is also shown that for every p q d + 1 there is a C = C(p, q, d) < such
that for every family G of compact convex sets in Rd
so that among any p of them some q have
a common hyperplane transversal, there is a set of at most C hyperplanes that together meet
all the members of G.
1 Introduction
In this paper we study geometric problems of the type introduced in [14] and considered in various
subsequent papers. It is convenient, however, to make the required definitions in the more general
framework of abstract families of sets. Let H be a (finite or infinite) family of (finite or infinite)
sets, and let F be another family of sets. For two integers p q we say that H satisfies the (p, q)
