How Sharp is the Concentration of the Chromatic Number?

Extract

As usual, let us write $G_{n,p}$ for a random graph with vertex set $[n]=\{1, 2, \dots , n\}$, in which the edges are chosen independently, with probability $p$. Similarly, $G_{n,m}$ is a random graph on $[n]$ with $m$ edges. For $p=m/({{n}\atop{2}})$, in many respects, the random graphs $G_{n,m}$ and $G_{n,p}$ are practically indistinguishable. (See [4] for an introduction to random graphs.) When in the late 1950s and early 1960s Erdős and Rényi founded the theory of random graphs, one of the most important problems they left open was the determination of the chromatic number of a random graph $G_{n,m}$. The original question concerned the case $m=O(n)$, but when in the 1970s Erdős popularized the problem, he was asking for good estimates on the chromatic number of $G_{n,m}$ for $m \sim n^2/4$ (equivalently, the chromatic number of $G_{{n,1/2}}$).