Two-Source Dispersers For Polylogarithmic Entropy And Improved Ramsey Graphs

In his 1947 paper that inaugurated the probabilistic method, Erdos proved the existence of (2+o(1)) log n-Ramsey graphs on n vertices. Matching Erd}os's result with a constructive proof is considered a central problem in combinatorics and has gained significant attention in the literature. The state-of-the-art result was obtained in the celebrated paper by Barak et al. [Ann. of Math. (2), 176 (2012), pp. 1483{1543], who constructed a 2(2(log log n)1-alpha) Ramsey graph for some universal constant alpha > 0. In this work, we significantly improve the result of Barak et al. and construct 2((log log n)c) -Ramsey graphs, for some universal constant c. In the language of theoretical computer science, this resolves the problem of explicitly constructing dispersers for two n-bit sources with entropy polylog(n). In fact, our disperser is a zero-error disperser that outputs a constant fraction of the entropy. Previously, such dispersers could only support entropy Omega(n).