The last remaining open problem from Erdős and Rényi's original paper on random graphs is the following: for q at least 3, what is the largest d so that the random graph G(n,d/n) is q-colorable with high probability? A lot of interesting work in probabilistic combinatorics has gone into proving better and better bounds on this q-coloring threshold, but the full answer remains elusive. However, a non-rigorous method from the statistical physics of glasses - the cavity method - gives a precise prediction for the threshold. I will give an introduction to the cavity method, with random graph coloring as the running example, and describe recent progress in making parts of the method rigorous, emphasizing the role played by tools from extremal combinatorics. Based on joint work with Amin Coja-Oghlan, Florent Krzakala, and Lenka Zdeborová.
- Combinatorial Theory Seminar