Given a subgraph $H$ of a general Cayley graph Cay($G$,$S$), is there a bound on the number of edges in H dependent only on the sizes $|H|, |S|,$ and $|G|$?
As an example, consider the graph $H_n=Cay(S(2,n),C)$, where S(2,n) is the hyperoctahedral group, and $C$ is the generator set consisting of cycles (when $S(2,n)$ is viewed as the signed symmetric group). I am interested in finding subgraphs of $H_n$ which are isomorphic to the Birkhoff graph $B_n=Cay(S_n,C)$. I was surprised to learn for n=3 that no such subgraph exists due to the fact that any subgraph of size $6$ in $H_3$ has at most $9$ edges (rather than the requisite $15$). I would like to generalize this result, but have not had much success, and the above question seems like the most natural way to approach it.