Upper Tails for Cliques

With ${\xi }_k={\xi }_k^{n,p}$ the number of copies of $K_k$ in the usual (Erd\H{o}s-R'enyi) random graph $G(n,p)$, $p\ge n^{-2\mathrm{/}(k-1)}$ and $\eta >0$, we show when $k>1$ $$\mathrm{Pr}({\xi }_k>(1+\eta )\text{E}{\xi }_k)<\exp [-{\text{gO}}_{\eta ,k}\min \{n^2{p}^{k-1}\log (1\mathrm{/}p),n^k{p}^{\left(\genfrac{}{}{0px}{}{k}{2}\right)}\}]\mathrm{.}$$ This is tight up to the value of the constant in the exponent.