The VC-dimension of random subsets of finite groups

For a random subset of a finite group $G$ of cardinality $N$, we consider the VC-dimension of the family of its translates (equivalently the VC-dimension of a random Cayley graph) and prove a law of large numbers as $N\to \mathrm{\infty }$. This answers a question of McDonald–Sahay–Wyman.