  Green showed that there exist constants $C_1,C_2>0$ such that the clique
number $\omega$ of the random Cayley graph on $\mathbb{F}_2^n$ satisfies
$\lim_{n\to\infty}\mathbb{P}(C_1n\log n < \omega < C_2n\log n)=1$. In this
paper we find the best possible $C_1$ and $C_2$. Moreover, we prove that for
$n$ in a set of density $1$, clique number is actually concentrated on a single
value. As a simple consequence of these results, we also prove the one-point
concentration result for the chromatic number, thus proving the
$\mathbb{F}_2^n$ analogue of the famous conjecture by Bollob\'{a}s and giving
almost the complete answer to the question by Green.
