  Let ${H}=(V, {E})$ be a hypergraph on the vertex set $V$ and edge set
${E}\subseteq 2^V$. We show that number of distinct {\it traces} on any $k-$
subset of $V$, is most $k.{\hat \alpha}(H)$, where ${\hat \alpha}(H)$ is the
{\it degeneracy} of $H$. The result significantly improves/generalizes some of
related results. For instance, the $vc$ dimension $H$ (or $vc(H)$) is shown to
be at most $\log({\hat \alpha}(H))+1$ which was not known before. As a
consequence $vc(H)$ can be computed in computed in $n^{O( {\rm log}({\hat
\delta}(H)))}$ time. When applied to the neighborhood systems of a graphs
excluding a fixed minor, it reduces the known linear upper bound on the $VC$
dimension to a logarithmic one, in the size of the minor. When applied to the
location domination and identifying code numbers of any $n$ vertex graph $G$,
one gets the new lower bound of $\Omega(n/({\hat \alpha}(G))$, where ${\hat
\alpha}(G)$ is the degeneracy of $G$.
