Bounds on the diameter of Cayley graphs of the symmetric group
  In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points. 
John Bamberg, Nick Gill, Thomas Hayes, Harald Helfgott, \'Akos Seress,   Pablo Spiga
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