 Fixed-size clique appearance threshold

For fixed $k\ge 2$, a binomial random graph has no $k$-cliques with probability tending to one if ${\mu }_n\to 0$, and has at least one with probability tending to one if ${\mu }_n\to \infty$. The first assertion uses the first moment method. The second uses the overlap formula for the variance of a clique count: overlapping $k$-sets with $l\ge 2$ contribute $O(n^{-l}{p}^{-\left(\genfrac{}{}{0ex}{}{l}{2}\right)})$, and each contribution tends to zero above this threshold. The fixed-size qualification matters.