Is the Manickam-Miklós-Singhi Conjecture solved? This arXiv paper is claimed to contain a proof for the MMS conjecture. But it seems that this manuscript is not yet peer reviewed by other mathematicians. I personally tried to follow the paper, but after some point I couldn't due to the lack of explanation, or my ability to understand. Given the interest of mathematicians for this problem, I think the manuscript could have been noticed by mathematicians' community. What is the status of this paper?
 A: The paper has appeared in print:
http://link.springer.com/article/10.1134%2FS0032946014040048
It does look strange that the author keeps posting and posting solutions to very famous problems in combinatorics every few months or so (yesterday a paper appeared claiming to have solved the famous conjecture on the singularity probability of a random Bernoulli matrix) and they all appear in 1 journal or do not appear at all. As the length of papers is 10-15 pages and they claim to solve long standing open problems, I would expect the refereeing process to be lightning fast and the best journals trying to snap those papers as soon as possible. Yet it does not happen.
The papers are genuinely terribly written (I have looked at them all quite a bit as I am interested) and no effort seems to have been put in making the results more readable. It is therefore strange that the author keeps posting and posting new supposed breakthroughts without making the previous ones more accessible (if the proofs are correct, hordes of combinatorialists and probabilists would be lining up to read it and present it in seminar, but we have absolute silence instead). 
But this is only my opinion. I would be very glad if my gloomy outlook is misplaced.
A: I am aware of the paper, but I am not sure that MO is the right forum for this sort of question.  Nonetheless, let me try to provide some information in as neutral a manner as possible.  
Note that there has been a flurry of recent activity concerning the MMS conjecture.  Indeed, a paper of Huang and Sudakov just appeared in the Electronic Journal of Combinatorics.  Their main result implies a cubic bound for MMS and it also resolves the vector space analogue of MMS.  There is also a very recent paper of Chowdhury, Sarkis and Shahriari which has better bounds than the Huang and Sudakov paper.  This paper was recently accepted by JCTa, and will appear soon.  Finally, there is a paper by Prokovskiy which proves that the MMS conjecture holds when $n \geq 10^{46}k$.  As far as I know, this paper is still under review and has not been accepted yet. This summary is by no means comprehensive. 
Regarding the Blinovsky paper, the actual strategy is to prove a stronger statement, namely the Ahlswede-Khachatrian conjecture (up to a finite number of exceptions).  The remaining cases can then be checked by computer.  An earlier paper by Aydinian and Blinovsky proves that MMS does indeed follow from Ahlswede-Khachatrian.  This paper has already appeared.    
Finally, note that the Blinovsky paper is not cited in the above Huang and Sudakov paper, and that in the Chowdhury, Sarkis and Shahriari paper, MMS is still listed as an open problem.  Of course this does not mean the paper is necessarily incorrect.  I think we will just have to wait and let the peer-review process play itself out.  
A: Yes, this paper is still under review, but I have no any reason that there are
any mathematical gap in the proof. Moreover there is even father developing
of the method, offered in this work, which allows to prove two old (50 years) problems: Erdos Matching conjecture and $s$-wise $t$-intersecting problem (see my Arxiv). The last
results have already passed the review and the reason why it passed   before
(as I was informed)
is that MMS paper contains some concrete calculations which is necessary to check carefully. 
About how to come from exponentials to exponents, simplest example:
if x^m +x^n =x^p +x^q for  sufficiently small $x$ and positive $m,n,p,q$,
then we have $m=p,n=q$ or vice versa. In the paper $x=e^{-1/\sigma^2}$ and $\sigma$
can be chosen arbitrary small. Also I should mention that I hope soon I will update both papers to the final variant, they will contain review and editor corrections, important for pleasure reading.  
A: I am happy to see that many mathematicians are interested in my conjecture. I derived this conjecture while I was working on "Distribution Invariants of Association Schemes" for my ph.d work at The Ohio State University in the mid 80's.It has been thirty years and from what I know the conjecture is not fully settled.
