= Solution
Realize the random <graph> process by giving every edge an independent uniform arrival time. For a fixed four-set $S$, only its six internal edges and its $4(n-4)$ outgoing edges affect whether it is an isolated <clique>. It is an isolated <clique> at some time exactly when all six internal edges arrive before every outgoing edge.
The relative ordering of these $4n-10$ edges is uniform. The chance that the first six are exactly the six internal edges is $\binom{4n-10}{6}^{-1}$. A <union bound> over the possible vertex sets gives
$$
\boxed{P(\text{an isolated four-clique ever appears})\le\frac{\binom n4}{\binom{4n-10}{6}}=O(n^{-2})\longrightarrow0}.
$$
Edges entirely outside $S$ are irrelevant. Counting the whole arrival-order event avoids an invalid union over continuously many times.
This is an example of <isolated clique appearance in a uniform random graph process>.
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