Let $n \in \mathbb N$ and $\sigma,\tau \subset {\rm Sym}(n)$ be generators.

Question: What is the maximal possible girth (if one varies $\sigma, \tau$) of the associated Cayley graph?

I am interested in asymptotic bounds. In [A. Gamburd, S. Hoory, M. Shahshahani, A. Shalev, B. Virág, On the girth of random Cayley graphs. Random Structures Algorithms 35 (2009), no. 1, 100–117.] it was proved that a random pair will produce girth $\Omega( (n \log(n))^{1/2})$ with probability tending to $1$. However, I am not aware of concrete generators that realize this girth.

A natural guess for a lower bound on the maximal possible girth (and maybe even the typical girth) would be $\Omega(n \log(n))$. It is also clear that $O(n \log(n))$ is an upper bound for the maximal possible girth.

To the best of my knowledge, it is not known if ${\rm Sym}(n)$ does satisfy a law of length $O(n)$. Hence, any superlinear lower bound on the maximal possible girth would have an immediate application.

Let $n \in \mathbb N$ and $\{\sigma,\tau\} \subset {\rm Sym}(n)$ be a generating set.

Question: What is the maximal possible girth (if one varies $\sigma, \tau$) of the associated Cayley graph?

I am interested in asymptotic bounds. In [A. Gamburd, S. Hoory, M. Shahshahani, A. Shalev, B. Virág, On the girth of random Cayley graphs. Random Structures Algorithms 35 (2009), no. 1, 100–117.] it was proved that a random pair will produce girth $\Omega( (n \log(n))^{1/2})$ with probability tending to $1$. However, I am not aware of concrete generators that realize this girth.

A natural guess for a lower bound on the maximal possible girth (and maybe even the typical girth) would be $\Omega(n \log(n))$. It is also clear that $O(n \log(n))$ is an upper bound for the maximal possible girth.

To the best of my knowledge, it is not known if ${\rm Sym}(n)$ does satisfy a law of length $O(n)$. Hence, any superlinear lower bound on the maximal possible girth would have an immediate application.