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We give a combinatorial proof of the following theorem. Let G be any finite group acting transitively on a set of cardinality n. If S G is a random set of size k, with k (n) ^1+ for some >0, then the diameter of the corresponding Schreier graph is O (ₖ n) with high probability. Except for the implicit constant, this result is the best possible.
Dona et al. (Mon,) studied this question.