Let $S\subset S_n$ be the set of all $n$-cycles. I want to know if the Cayley graph $(S_n,S)$ has large dense subgraphs. I'm expecting it to not have super-polynomial size and $1-o(1)$ dense subgraphs.

However, one good starting point maybe the following question:

What is the size of the largest clique (and biclique) of $(S_n,S)$?

As $(S_n,S)$ is a subgraph of the Birkhoff polytope graph $B_n$, a bound on the clique size of $B_n$ might be also helpful. I find this question closely related, but no bounds in terms of $n$ were given there.

Another related result I find is on the independence number of $B_n$, proved in Kane, Lovett and Rao:

The independence number $\alpha(B_n)$ satisfies $n!/4^n\leq\alpha(B_n)\leq n!/2^{(n-4)/2}$.