SIAM Journal on Computing, Ahead of Print.
In his 1947 paper that inaugurated the probabilistic method, Erdös proved the existence of $(2+o(1))\log{n}$-Ramsey graphs on $n$ vertices. Matching Erdös'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 $\Omega(n)$.

中文翻译：

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用于多对数熵和改进的Ramsey图的两源分散器

《 SIAM计算杂志》，预印本。
Erdös在其1947年提出概率方法的论文中证明了$ n $顶点上存在$（2 + o（1））\ log {n} $-Ramsey图。将Erdös的结果与建设性的证明相匹配被认为是组合学的中心问题，并且在文献中也引起了极大的关注。最新的结果是Barak等人在著名论文中获得的。[安 数学。（2），176（2012），pp。1483--1543]，他为某些对象构造了$ 2 ^ {2 ^ {（\ log \ log {n}）^ {1- \ alpha}}} $-Ramsey图通用常数$ \ alpha> 0 $。在这项工作中，我们显着改善了Barak等人的结果。并为某些通用常数$ c $构造$ 2 ^ {（\ log \ log {n}）^ c} $-Ramsey图。用理论计算机科学的语言，这解决了显式构造两个具有熵$ {{{polylog}}（n）$的$ n $位源的分散器的问题。实际上，我们的分散器是零误差分散器，它输出恒定比例的熵。以前，此类分散器只能支持熵\\ Omega（n）$。