I will discuss a generalization of Friedman's theorem, focusing on the spectral gap of the random regular Schreier graphs associated with the action of S_n on K_n--tuples of distinct elements in {1,...,n}. The proof relies on the `polynomial method', a new approach to strong convergence of Chen, Garza--Vargas, Tropp and van Handel, combined with new group theoretic inputs. A key ingredient is a new asymptotic bound on the expected character of a random permutation obtained via a word map, which is expressed in terms of the dimension of the corresponding representation.
