The diameter of random Schreier graphs

We give a combinatorial proof of the following theorem. Let $G$ be any finite group acting transitively on a set of cardinality $n$. If $S\subseteq G$ is a random set of size $k$, with $k\ge {(logn)}^{1+\varepsilon }$ for some $\varepsilon >0$, then the diameter of the corresponding Schreier graph is $O\left({log}_{k}n\right)$ with high probability. Except for the implicit constant, this result is the best possible.