= Isolated clique appearance in a uniform random graph process
In a <uniform random graph process>, realize edge arrivals by independent continuous uniform times. For a fixed set of $r$ vertices to form an isolated <clique> at some time, all $e=\binom r2$ internal edges must arrive before the first of the $r(n-r)$ outgoing edges. Their relative arrival ordering is uniform, so the exact probability is
$$
\binom{r(n-r)+e}{e}^{-1}.
$$
A <union bound> over the vertex sets therefore bounds the probability that such an isolated clique ever appears by $\binom nr/\binom{r(n-r)+e}{e}$. For fixed $r\ge4$ this is $O(n^{r-e})\to0$, since $e>r$. This argument controls all process times at once, including the short interval during which an isolated component could exist, without discretizing time.
Back to article page
