Theorems about piercing numbers I'm searching for results relating to piercing numbers. One example of what I'm looking for is this theorem: any VC Class which is k-consistent has a bounded piercing number.
However, searching Google/arxiv only gives me the above theorem and a bunch of papers about convex sets. What are some other results/papers related to piercing numbers? Ideally, they should be combinatorial in nature as opposed to geometric.
Background: I'm asking this because I'm currently doing research on the possible connection between VC Classes and compression schemes.
 A: One of the most general results is that of Alon and Kalai in their 1995
paper "Bounded the piercing number,"
solving the 1957 "$(p,q)$ problem" of Hadwiger and Debrunner.
The show that, if there is a family of sets $\cal F$
(condition on these sets later) so that any $p$ of them
contain a subset of $q$ with a non-empty intersection, $p \ge q$,
then there is a set of at most $c$ points
that intersects each member of $\cal F$, i.e., $c$ points pierce $\cal F$.
The condition on $\cal F$ is that each member is the union of at most $k$ compact,
convex sets in $\mathbb{R}^d$.  Of course $c$ depends on all the parameters
$\lbrace p, q, k, d \rbrace$, but the important point is that $c$ is finite.
This is a geometric result, rather than a purely combinatorial one, but
it is very broad.
Discrete and Computational Geometry, Vol. 13, No. 1, 245-236, 1995.
You might also look at "the Colorful Caratheodory Theorem," described, e.g., at Gil Kalai's blog.
A: The result you are looking for is proved in Matousek's paper "bounded VC-dimension implies a fractional Helly theorem". It is theorem 4 there.
Matousek explains how to adapt Alon and Kleitman's proof of the $(p,q)$-theorem mentioned in Joseph's answer from families of convex sets to families of finite VC-dimension.
