We prove that every constant a>1/2 the chromatic number of the random graph G(n,p) with p=n^{-a} is almost surely concentrated in two consecutive values. This implies that for any b<1/2 and any integer valued function r(n)=O(n^b) there exists a function p(n) such that the chromatic number of G(n,p(n))is precisely(!) r(n) almost surely.
This is a joint work with Noga Alon.