Written: Jan. 22, 2018
One of the hardest parts of combinatorics is Ramsey theory, nicely covered by the Graham-Rotschild-Spencer classic and the more recent book by Landman and Robertson. We have no clue what R(5) is, and possibly never will (at any rate, we will [most probably] never know the value of R(6)). Also the gap between the proved upper bound for R(n) and the proved lower bound is huge! (roughly 4n and sqrt(2)n respectively). Hence we desperately need new ideas and new methodologies, that will increase our knowledge , not necessarily by rigorous proofs.
A fairly recent breakthrough by mathematician Aaron Robertson, statistician William Cipolli, and undergraduate student Maria Dascalu, does just that. The paper speaks for itself, and I hope that you will read it. While non-rigorous, it is more certain than many believed-to-be-proved results in mathematics, that often contain (as yet undetected) both minor and major gaps.
Aaron gave a great talk about it in the recent JMM in San Diego, and I was shocked when he told me, that after he submitted it to the Proc. of the National Academy of Science (PNAS), it was immediately rejected, without being sent for review. A form letter, signed by the (at the time) editor-in-chief, Inder Verma, stated that "the expert that served as an editor" decided that it is of "insufficient general interest" and should go to a specialized journal. Unfortunately, he did not specify the name of that "expert editor", but the website of PNAS only lists three mathematical editors: Ken Ribet, Rob Kirby, and Srinivasa Varadhan. When I asked Ken Ribet to reconsider the poor decision, he first stated that "he never saw it", so unless Professor Verma was mistaken, this narrows down the culprits to either Kirby or Varadhan.
Whoever made this decision, made a very poor one. It shows the dismissive attitude of traditional mathematicians towards experimental and innovative mathematics. It is just as bad as a dismissive attitude towards women and minorities. I also have another conjecture. They saw "Colgate University" and immediately dismissed it as of "insufficient general interest", in other words they were dismissive of small universities. Finally, I was also disappointed at Ron Graham, who politely stated that the "final decision" is by the editors (e.g. Ken Ribet) and there is nothing he can do. Since members of the NAS are allowed to submit papers, he should have tried to over-rule the poor editorial decision.
Ken Ribet, in the above-mentioned email, stated that "the editorial process is very noisy", so the best thing to do, according to him, when a paper gets rejected, rather than to try and appeal it, is to try another journal. While this sounds like sound advice, I must say that the "noise" is not exactly symmetric, it is a very biased noise.
Luckily, nowadays we have arxiv.org, so being accepted by a "real" journal is not that important. The future will decide that this work is more interesting and more significant that most of the mathematical articles that did get accepted by the PNAS.