This is a fantastic result!
It would have been great to see any explicit two-source extractor for min-entropy rate below one half, let alone one that beats Bourgain's rate of 0.499. Handling any constant min-entropy rate would have been a feast (see A Challenge from the mid-1980s), and going beyond that would have justified a night-long party. A next challenge could have been handling min-entropy $n^c$, for any constant $c>0$. But the current work ``saves'' us all these intermediate celebrations by presenting us with a construction that handles poly-logarithmic min-entropy.
Yes, one can be greedy and seek to extract more bits and/or obtain a negligible extraction error. But let's party today...
The main theorem is proved by combining two (non-standard types of) extractors.
Denoting the latter extractor (for non-oblivious bit-fixing sources) by bfEXT,
and using an ordinary extractor EXT that uses a seed of length $d=O(log n)$
and extracts $d'$ bits, we obtain the two-source extractor that,
on input $(x,y)\in(\{0,1\}^n)^2$, proceeds as follows:
1. For every $d$-bit long seed $s$, produces $z_s=nmEXT(y,E(x,s))$;
2. letting $z = z_{0^d}\cdots z_{1^d}$, it outputs bfEXT(z).
N.B.: Using a polylog independence parameter for nmEXT,
requires having a seed of super-logarithmic seed.
The standrad extractor EXT is used to extract such a seed from
the first source X, and this is used to sample seeds of the nmEXT.
We explicitly construct an extractor for two independent sources on $n$ bits, each with min-entropy at least $\log^C n$ for a large enough constant $C$. Our extractor outputs one bit and has error $n^{-\Omega(1)}$. The best previous extractor, by Bourgain [B2], 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 $n^{1-\delta}$, 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 $polylog(n)$-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 \cite{Viola14}, achieved $q=n^{1/2 - \delta}$.
Our other main contribution is a reduction showing how such a resilient function gives a two-source extractor. This relies heavily on the new non-malleable extractor of Chattopadhyay, Goyal and Li [CGL15].
Our explicit two-source extractor directly implies an explicit construction of a $2^{(\log \log N)^{O(1)}}$-Ramsey graph over $N$ vertices, improving bounds obtained by Barak et al. [BRSW12] and matching independent work by Cohen [Coh15b].
See ECCC TR15-119.