THE CONCENTRATION OF THE CHROMATIC NUMBER OF RANDOM GRAPHS

We prove that for every constant delta > 0 the chromatic number of the random graph G(n,p) with p = n(-1/2-delta) is asymptotically almost s urely concentrated in two consecutive values. This implies that for an y beta < 1/2 and any integer valued function r(n) less than or equal t o O(n(beta)) there exists a function p(n) such that the chromatic numb er of G(n,p(n)) is precisely r(n) asymptotically almost surely.