One of the earliest and best-known application of the probabilistic method is the proof of existence of a 2 log n-Ramsey graph, i.e., a graph with n nodes that contains no clique or independent set of size 2 log n. The explicit construction of such a graph is a major open problem. We show that a reasonable hardness assumption implies that in polynomial time one can construct a list containing polylog(n) graphs such that most of them are 2 log n-Ramsey