  In this work we show that given a connectivity graph $G$ of a $[[n,k,d]]$
quantum code, there exists $\{K_i\}_i, K_i \subset G$, such that $\sum_i
|K_i|\in \Omega(k), \ |K_i| \in \Omega(d)$, and the $K_i$'s are
$\tilde{\Omega}( \sqrt{{k}/{n}})$-expander. If the codes are classical we show
instead that the $K_i$'s are $\tilde{\Omega}\left({{k}/{n}}\right)$-expander.
We also show converses to these bounds. In particular, we show that the BPT
bound for classical codes is tight in all Euclidean dimensions. Finally, we
prove structural theorems for graphs with no "dense" subgraphs which might be
of independent interest.
