Random obstruction to larger clique blow-ups ($\mathbb{E}\#K_s(t)\le n^{st}{p}^{\left(\genfrac{}{}{0ex}{}{s}{2}\right)t^2}$)

In a binomial random graph with fixed edge probability below one, the displayed first-moment bound tends to zero for $t=C\log n$ and sufficiently large fixed $C$. At the same time the edge density concentrates near $p$. Thus a positive fixed density cannot force balanced complete multipartite subgraphs whose part size grows faster than logarithmically.