In his 1947 paper that inaugurated the probabilistic method, Erdős proved the existence of $2 \log(n)$-Ramsey graphs on $n$ vertices. Matching Erdős' result with a constructive proof is a central problem in combinatorics that has gained a significant attention in the literature. In this talk we will present a recent work towards this goal (http://eccc.hpi-web.de/report/2015/095/).