Let $G$ be a random graph on $n$ vertices, in which every edge is included independently with probability $1/2$. Is there some constant $C$ such that that chromatic number $\chi (G)$ is, almost surely, concentrated on at most $C$ values? Is it true that, if $\omega (n)\to \infty$ sufficiently slowly, then for every function $f(n)$$$\mathbb{P}(|\chi (G)-f(n)|<\omega (n))<1/2$$if $n$ is sufficiently large?