= Solution
The requested scale is $\boxed{k=\Theta(\sqrt{\log n})}$. To see its first-moment origin, the number $X_k$ of $k$-cliques has <expectation>
$$
\mathbb E X_k=\binom nk2^{-\binom k3}.
$$
For $k$ on the proposed scale, taking base-two <logarithms> gives
$$
\log_2\mathbb E X_k=k\log_2 n-\frac{k^3}6+O(k^2+k\log k).
$$
The leading terms balance at $k\approx\sqrt{6\log_2 n}$. In particular, for any fixed $c>\sqrt6$, taking $k$ near $c\sqrt{\log_2 n}$ makes the <expectation> tend to zero, so such <cliques> are absent with <probability> tending to one.
Thus the first-moment estimate places the largest-clique scale at a <constant> multiple of $\sqrt{\log n}$, as requested. A diverging <expectation> below that balance does not itself prove existence; a corresponding probabilistic lower-bound proof would require an additional argument and is not part of the requested first-moment estimate.
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