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 But it seems that this manuscript is not yet peer reviewed by other mathematicians.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?

3$\begingroup$ I'm a little suspicious of the claim made on page 6, which seems to be that if two sums of exponentials are equal then the two corresponding sets of exponents are also equal. $\endgroup$– zebJul 19, 2014 at 15:18

1$\begingroup$ The paper has appeared link.springer.com/article/10.1134%2FS0032946014040048 $\endgroup$– Thomas KalinowskiJan 23, 2015 at 3:05

4$\begingroup$ I'm voting to close this question as offtopic because it asks about the validity of a preprint/paper. See meta.mathoverflow.net/questions/2328/… and meta.mathoverflow.net/questions/927/… for the relevant policy. $\endgroup$– user9072Nov 20, 2015 at 17:18

4$\begingroup$ So to answer the last question: the status of the paper is that it has been published, as mentioned by Thomas Kalinowski. Otherwise, this question has problems. It's quite all right to ask about the validity of a specific claim or argument in the paper: where in the paper does the OP get stuck, exactly? But this question is too vague and "primarily opinionbased", and invites answers based on circumstantial evidence (the accepted answer gives context, but the mathematics is indirect). Thus, I have to vote to close. $\endgroup$– Todd Trimble ♦Nov 20, 2015 at 17:38
4 Answers
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 AhlswedeKhachatrian 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 AhlswedeKhachatrian. 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 peerreview process play itself out.

$\begingroup$ I think a more neutral and direct answer would be along the lines of "yes, I have read the paper", or "no, I haven't", or "I talked to someone who said she had read the paper". Although I appreciate the background in your post, it does not address what I think the real question is: "Does anyone know of any independent verification or refutation of (ideas in) paper X?". Then those who have such information can decide to share that here on MathOverflow (with opinions or statements about the correctness taken elsewhere, not here), or not. Gerhard "Doesn't Know Who Read It" Paseman, 2014.07.19 $\endgroup$ Jul 19, 2014 at 18:58

$\begingroup$ Point taken. However, my intention was to be neutral and thus necessarily indirect. For what it is worth, I do know someone who attempted to read the paper, but I think sharing that opinion is just hearsay. Since I have not read the paper carefully myself, I don't think I should say anymore. $\endgroup$ Jul 19, 2014 at 21:05

$\begingroup$ Thank you. Your last sentence in your comment would make an excellent ending to your post. $\endgroup$ Jul 19, 2014 at 23:35

1$\begingroup$ There is an improvement on the result mentioned in the paper of A. Chowdhury which is mentioned here: arxiv.org/abs/1405.0909 $\endgroup$– AnuragSep 17, 2014 at 17:18
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 1015 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.

1$\begingroup$ Oh yes, the author has recently posted the paper claiming to solve yet another whalesized problem  the union closed conjecture. Either the man is a genius and the world of mathematics should celebrate him, or maybe more like a fermatist every month or so sending a new proof of some long standing conjecture. Anyhow, a conlusive diagnosis would be handy. $\endgroup$– TOMNov 18, 2015 at 12:26

4$\begingroup$ I downvote, and flag this post for moderator's attention as it is a speculation about the papers done behind a veil of anonymity. If you have a particular problem with the paper, it is OK to mention it, but this kind of nonmathematical talk is not constructive. Saying rumours "Yet I know..." without sources is especially damaging. Are you the editor, are you the referee? If so, stand behind your decisions, and reveal your identity and specific concerns about the paper. Otherwise, how do you know? $\endgroup$ Nov 20, 2015 at 16:38

2$\begingroup$ I agree that any factual assertions ought to be sourced, and this answer should either undergo a major edit to remove innuendo, or be deleted. I am going to delete the first sentence particularly as it seems to impute bad faith to the paper's author without any sort of documentation. Virtually the entirety of the post looks like expression of opinion and innuendo without any discussion of what might be wrong with the mathematics. $\endgroup$– Todd Trimble ♦Nov 20, 2015 at 17:35

$\begingroup$ Thank you sharing your concerns. I will delete all rumorlike content from the post that I cannot back up by revealing the source. $\endgroup$– TOMNov 21, 2015 at 22:16

$\begingroup$ It is hard to answer the question without speculation, so instead of putting the blame on this answer, blame the question. (Which now has been closed.) $\endgroup$– domotorpJan 22, 2016 at 3:42
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.

1$\begingroup$ The thing that makes me suspicious here is that $m,n,p,q$ seem to be functions of $x$, not constants. In this case I think the only thing one can really conclude is that the minimum of $m,n$ is close to the minimum of $p,q$. For instance, I don't see what rules out situations like $100^{1} + 100^{10} = 100^{1.01} + 100^{1.67335...}$. $\endgroup$– zebAug 1, 2014 at 23:06

$\begingroup$ No, they are positive constants independent of x. It is important that x is sufficiently small, for given value it can be not true. If minimums are equal, you can skip these exponentials and continue. But I just come across the discussion in this blog and reply. Not excellent writing paper is disadvantage but when famous problem is solved not just regular research is made, I think the most important thing is whether the proof is right modulo some corrections or not $\endgroup$– MikhailAug 1, 2014 at 23:23

$\begingroup$ Better example: we can take $x$ very small, $m=1,n=10$, $p=1+x$, and $q = 2\log(\log(1/x))/\log(1/x) + O(x)$. $\endgroup$– zebAug 2, 2014 at 0:18

7$\begingroup$ Mikhail: The paper is either correct or it is not, there can be no modulo. And you are indeed correct  this is not just a problem  it is a major problem in the field and so the standards should be higher. It is not the job of the referee to sit down and work on the details for weeks. No one has to do this work but the author or some person he could hire. $\endgroup$– TOMNov 26, 2015 at 21:04
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.

7$\begingroup$ Welcome to MathOverflow! Can you say anything to resolve the question as asked? Following your last sentence, the preferred answers would be something like "I have not fully read the ArXiv paper." or "I have read it and have an issue with point X raised on page Y. The issue with the mathematics is Z. This and other points need to be resolved before I consider the conjecture fully settled." Gerhard "Happy To Talk Math Here" Paseman, 2015.10.08 $\endgroup$ Oct 8, 2015 at 16:12