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Consider the Cayley graph of $S_n$ generated by a random pair of elements$x,y$. Conjecturally, the girth of this graph is $\Omega(n \log n)$ withprobability tending to $1$ as $n\to\infty$. We show that it is at least$\Omega(n^{1/3})$.

The trivial lower bound for the girth of $S_n$