TITLE: Long gaps between primes
QUESTION [8 upvotes]: Zhang's celebrated result established the existence of bounded gaps between primes, that is, there exists a constant $B$ (in Zhang's paper $B$ can be taken to be $7 \times 10^7$, and this was almost immediately improved significantly so that one may take $B= 4680$) such that there are infinitely many primes $p_n$ such that $p_n - p_{n-1} < B$. 
My question considers the opposite case. Do there exist, infinitely often, gaps that are much larger than average? The prime number theorem implies that the average gap between $p_n$ and $p_{n+1}$ is about $\log p_n$.
Thus, for which constant $C > 1$ can one establish the existence of infinitely many primes $p_n$ such that $p_n - p_{n-1} > C\log p_n$? And what is the best known constant to date?
Edit: in view of the wikipedia entry on this topic, it seems that the correct thing to look at is
$$\displaystyle p_n - p_{n-1} > \frac{c \log n \log \log n \log \log \log \log n}{(\log \log \log n)^2}.$$
This seems like a rather unnatural function, but for now it is not known if the constant $c$ in the above inequality may be taken to be arbitrarily large. Is this inequality expected to be the right order of magnitude?
Thanks for any insight.

REPLY [4 votes]: Erdos's problem, which asks whether or not the bound
$$\displaystyle p_n - p_{n-1} > \frac{c \log n \log \log n \log \log \log \log n}{(\log \log \log n)^2}$$
holds with $c$ arbitrarily large, has been answered in the positive independently by Ben Green, Kevin Ford, Sergei Konyagin, and Terry Tao and James Maynard. However, it seems that they have decided to write a single paper on the subject, which is found here: http://arxiv.org/abs/1412.5029