Explicit Two-Source Extractors and Resilient FunctionQuoteAbstract:We explicitly construct an extractor for two independent sources on n bits, each with min-entropy at least logCn for a large enough constant~C. Our extractor outputs one bit and has error n−(1). The best previous extractor, by Bourgain, required each source to have min-entropy 499n.A key ingredient in our construction is an explicit construction of a monotone, almost-balanced boolean function on n bits that is resilient to coalitions of size n1−, for any 0. In fact, our construction is stronger in that it gives an explicit extractor for a generalization of non-oblivious bit-fixing sources on n bits, where some unknown n−q bits are chosen almost \polylog(n)-wise independently, and the remaining q=n1− bits are chosen by an adversary as an arbitrary function of the n−q bits. The best previous construction, by Viola, achieved q=n12− .Our explicit two-source extractor directly implies an explicit construction of a 2(loglogN)O(1)-Ramsey graph over N vertices,improving bounds obtained by Barak et al. and matching independent work by Cohen.