2-source dispersers for $n^{o(1)}$ entropy, and Ramsey graphs beating the Frankl-Wilson construction

Abstract

The main result of this paper is an explicit disperser for two independent sources on $n$ bits, each of min-entropy $k=2^{\log^{\beta} n}$, where $\beta\lt 1$ is some absolute constant. Put differently, setting $N=2^n$ and $K=2^k$, we construct an explicit $N \times N$ Boolean matrix for which no $K \times K$ sub-matrix is monochromatic. Viewed as the adjacency matrix of a bipartite graph, this gives an explicit construction of a bipartite $K$-Ramsey graph of $2N$ vertices. This improves the previous bound of $k\!=\!o(n)$ of Barak, Kindler, Shaltiel, Sudakov and Wigderson. As a corollary, we get a construction of a $2^{n^{o(1)}}$ (nonbipartite) Ramsey graph of $2^n$ vertices, significantly improving the previous bound of $2^{\tilde{O}(\sqrt{n})}$ due to Frankl and Wilson. We also give a construction of a new independent sources extractor that can extract from a constant number of sources of polynomially small min-entropy with exponentially small error. This improves independent sources extractor of Rao, which only achieved polynomially small error. Our dispersers combine ideas and constructions from several previous works in the area together with some new ideas. In particular, we rely on the extractors of Raz and Bourgain as well as an improved version of the extractor of Rao. A key ingredient that allows us to beat the barrier of $k=\sqrt{n}$ is a new and more complicated variant of the challenge-response mechanism of Barak et al. that allows us to locate the min-entropy concentrations in a source of low min-entropy.