Random obstruction to larger clique blow-ups
= Random obstruction to larger clique blow-ups
{title2=$\mathbb E\#K_s(t)\le n^{st}p^{\binom s2t^2}$}
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.