Upper Tails for Cliques

Abstract

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 $G(n,p)$, $p\geq n^{-2/(k-1)}$ and $\eta>0$, we show when $k>1$ $$\Pr(\xi_k> (1+\eta)\E \xi_k) < \exp [-\gO_{\eta,k} \min\{n^2p^{k-1}\log(1/p), n^kp^{\binom{k}{2}}\}].$$ This is tight up to the value of the constant in the exponent.