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.



Theorems about piercing numbers
One of the most general results is that of Alon and Kalai in their 1995
paper "*Bounding 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. 3–4, 245–256, 1995](https://doi.org/10.1007/BF02574042 "EuDML:131360. zbMATH review at https://zbmath.org/0826.52006").


You might also look at "the Colorful Caratheodory Theorem," described, e.g., at [Gil Kalai's blog](https://gilkalai.wordpress.com/2009/03/15/colorful-caratheodory-revisited/).


