Abstract
We explicitly construct an extractor for two independent sources on n bits, each with min-entropy at least $\mathrm{log}^C n$ for a large enough constant $C$. Our extractor outputs one bit and has error $n^{-\Omega(1)}$. The best previous extractor, by Bourgain, required each source to have min-entropy $.499n$. M
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 $n^{1-\delta}$ for any $\delta>0$. 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 $\mathrm{polylog}(n)$-wise independently, and the remaining $q=n^{1-\delta}$ bits are chosen by an adversary as an arbitrary function of the $n-q$ bits. The best previous construction, by Viola, achieved $q=n^{1/2-\delta}$.
Our explicit two-source extractor directly implies an explicit construction of a $2^{(\mathrm{log}\ \mathrm{log}\ N)^{O(1)}}$-Ramsey graph over $N$ vertices, improving bounds obtained by Barak et al. and matching an independent work by Cohen.
Citation
Eshan Chattopadhyay. David Zuckerman. "Explicit two-source extractors and resilient functions." Ann. of Math. (2) 189 (3) 653 - 705, May 2019. https://doi.org/10.4007/annals.2019.189.3.1
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