  We consider random simple temporal graphs in which every edge of the complete
graph $K_n$ appears once within the time interval [0,1] independently and
uniformly at random. Our main result is a sharp threshold on the size of any
maximum $\delta$-clique (namely a clique with edges appearing at most $\delta$
apart within [0,1]) in random instances of this model, for any
constant~$\delta$. In particular, using the probabilistic method, we prove that
the size of a maximum $\delta$-clique is approximately
$\frac{2\log{n}}{\log{\frac{1}{\delta}}}$ with high probability (whp). What
seems surprising is that, even though the random simple temporal graph contains
$\Theta(n^2)$ overlapping $\delta$-windows, which (when viewed separately)
correspond to different random instances of the Erdos-Renyi random graphs
model, the size of the maximum $\delta$-clique in the former model and the
maximum clique size of the latter are approximately the same. Furthermore, we
show that the minimum interval containing a $\delta$-clique is
$\delta-o(\delta)$ whp. We use this result to show that any polynomial time
algorithm for $\delta$-TEMPORAL CLIQUE is unlikely to have very large
probability of success.
