Binomial random graph $G(n,p)$ is an $n$-vertex graph such that each of the $\binom{n}{2}$ pairs is included as an edge independently with probability $p$.
I am wondering with high probability (the probability of an event tends to 1 as $n$ tends to infinity), what is the size of the largest clique in $G(n,p)$? (Here $p$ might be a function of $n$, i.e., $p(n)$.)
Let $\omega_n$ denote the clique number of $G(n,p)$. For a constant $p$ Grimmett & McDiarmid showed that $$ \omega_n= -\frac{2}{\ln p}\ln n+o_{a.s.}(\ln n). $$