The Chromatic Number of Random Graphs for Most Average Degrees

For a fixed number d >0 and n large, let G(n,d/n) be the random graph on n vertices in which any two vertices are connected with probability d/n independently. The problem of determining the chromatic number of G(n,d/n) goes back to the famous 1960 article of Erdös and Rényi that started the theory of random graphs [Magayar Tud. Akad. Mat. Kutato Int. Kozl. 5 (1960) 17-61]. Progress culminated in the landmark paper of Achlioptas and Naor [Ann. Math. 162 (2005) 1333-1349], in which they calculate the chromatic number precisely for all d in a set S⊂(0,∞) of asymptotic density lim_z→∞ 1₀^z= 1_S = 1/2 , and up to an additive error of one for the remaining d. Here we obtain a near-complete answer by determining the chromatic number of G(n,d/n) for all din a set of asymptotic density 1.