  The well-known Sauer lemma states that a family $\mathcal{F}\subseteq
2^{[n]}$ of VC-dimension at most $d$ has size at most
$\sum_{i=0}^d\binom{n}{i}$. We obtain both random and explicit constructions to
prove that the corresponding saturation number, i.e., the size of the smallest
maximal family with VC-dimension $d\ge 2$, is at most $4^{d+1}$, and thus is
independent of $n$.
