  We prove a local limit theorem the number of $r$-cliques in $G(n,p)$ for
$p\in(0,1)$ and $r\ge 3$ fixed constants. Our bounds hold in both the
$\ell^\infty$ and $\ell^1$ metric. The main work of the paper is an estimate
for the characteristic function of this random variable. This is accomplished
by introducing a new technique for bounding the characteristic function of
constant degree polynomials in independent Bernoulli random variables, combined
with a decoupling argument.
