There is some news on this problem. There is a paper here: http://arxiv.org/pdf/1402.2184v1.pdf
It finds an example of size 1160 that has value 2 and proves that that example is maximal. The polymath 5 project found an example of size 1124. The proof that 1160 is maximal required a large computer proof. For more information see this post: http://gowers.wordpress.com/2014/02/11/recent-news-concerning-the-erdos-discrepancy-problem/
There is an uploaded version of 8a. Also there is going to be a article about the experience of working on 8a and possibly 8b for the newsletter of the European Mathematical society. The deadline is around April. For more check the post at http://terrytao.wordpress.com/2014/02/07/new-equidistribution-estimates-of-zhang-type-and-bounded-gaps-between-primes-and-a-retrospective/
Polymath 9 has answered one of the questions it asked apparently. The idea that was being tried didn’t work there was a counterexample. It could possibly continue if the method could be modified to avoid the counterexample. at http://gowers.wordpress.com/2014/01/09/dbd2-success-of-a-kind/ is a post discussing this. Meanwhile Polymath 8 has spawned a new project Polymath 8b. http://terrytao.wordpress.com/2014/01/17/polymath8b-vi-a-low-dimensional-variational-problem/ is the latest post of this project.
Polymath 9 has started. See this post: http://gowers.wordpress.com/2013/11/03/dbd1-initial-post/
Polymath 8 is working towards publication and it looks like Polymath 9 might be starting up. See the post: http://gowers.wordpress.com/2013/10/24/what-i-did-in-my-summer-holidays/
There is a new Polymath project proposal. It is about bounded gaps between primes. Here is the URL for the proposal:
http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/
There has been some progress in the polymath 7 project. See the new thread here.
