  Given natural numbers $k \leq s \leq n$, we ask: what is the minimal
VC-dimension of a family $\mathcal{F}$ of $s$-subsets of $[n]$ that covers all
$k$-subsets of $[n]$? We first show that for sufficiently large $n$ this number
is always $k$, and construct families which give a lower bound for the actual
growth of this stabilization point.
