The Polymath experiment is still very much in its infancy, with the result that we still have only a rather hazy idea of what the advantages and disadvantages are of open online multiple collaboration. It is easy to think of potential advantages and disadvantages, but the more actual experience we can draw on, the more we will get a feel for which of these plausible speculations are correct.
In my last post but one I outlined a few thoughts about the cap-set problem. Although this wasn’t meant as the beginning of a Polymath project (as I said in the post, it was modelled more on Gil Kalai’s notion of an “open discussion”), it had something in common with such projects, to the point where one of the worries people have about the Polymath approach actually occurred: I suggested a line of attack that was, unbeknownst to me, actively being pursued by somebody.
I do not know exactly what calculations went on in the minds of Michael Bateman and Nets Katz when they decided to go public with their ideas before they had fully worked them out, but my guess is that they wanted to establish that they had got there first. This they did by the very effective method of explaining that the simple observations that I had made were much weaker than what they already knew how to prove. As it happens, the story has had a happy ending for them, since they managed, soon after posting their comments, to push their ideas to the point where they have obtained the first improvement to the Roth/Meshulam bound, something that many people have wanted to do for a long time.
It’s possible therefore that the quasi-Polymathematical post of mine was actually beneficial to Bateman and Katz, providing them with just enough fear (almost certainly unwarranted, I should add) of getting scooped that they were stimulated into pushing their already excellent ideas through to their conclusion. I’d need to know their side of the story a bit better to know whether that really is the case, but at any rate I think it is fair to say that the outcome has been positive.
We now find ourselves in a very interesting situation. The argument of Bateman and Katz improves the Roth/Meshulam bound by a factor of for a very small absolute constant That is, the density they need is instead of the old Meanwhile, over in the context, there has been much excitement over the recent result of Sanders that improves the bound for Roth’s theorem to a density of We should think of as the equivalent of in the case, so is the equivalent of Thus, the Bateman-Katz bound would, if one could carry it over to , give a bound of and that gets people very excited because it would prove that every set of integers such that contains an arithmetic progression of length 3, which is the first non-trivial case of a very famous conjecture of Erdős.
Just in case the meaning of the title of this post is not obvious to everybody by this point, let me spell it out: let A be the the Roth/Meshulam proof, let B be the proof of Bateman and Katz and let C be Sanders’s proof. It is tempting to speculate that there exists a proof D that stands in relation to C as B does to A, allowing an improvement to Sanders’s bound by a factor of
The most interesting question concerning this speculation is of course whether it is correct. But the second most interesting question (well, perhaps I don’t really want to claim that, but any rate, an interesting question) is what is going to happen now. There are several people around with a strong interest in these problems and experience with similar questions. Should there now be a race, with all the glory going to the person, or small team of people, who can most quickly digest the existing arguments and find the right combination of them?
That would be the obvious non-Polymath way of doing things. The obvious Polymath way of doing things would be to do things the Polymath way — obviously. That is, everybody who wants to should participate in an attempt to understand exactly what is going on in the two recent papers (as well as certain other papers that have contributed to making these breakthroughs possible, notably papers of Croot and Sisask and of Katz and Koester) and to work out whether we do now have enough ingredients to break the barrier in Roth’s theorem.
The main purpose of this post is to say that a Polymath project of this kind is indeed going to start. This comes after quite a bit of emailing in the background, which has established that all of Sanders, Bateman and Katz are happy to do things this way (where I think “happy” means genuinely happy rather than agreeing-after-twisting-of-arm happy), as well as several other interested parties.
One of our concerns is that nobody who might be interested in participating should be excluded, so this is the moment where the private emailing stops and the conversation can take place in public. At the moment, the focus is on trying to understand the papers of Sanders and of Bateman and Katz, though there has been a small amount of assessment of what might need to be done, which I’ll discuss in a moment. An early priority is to set up a wiki, where people can add articles that contribute to this understanding. (I’m hoping, for instance, that people may write articles in which they try to explain what the ideas behind the various papers are: it would be interesting to have several people writing such articles, each with slightly different perspectives, even if the articles overlapped.) I have created a page for this purpose and added links to the articles of Bateman-Katz and Sanders. It can be reached by clicking here or by following a link in the sidebar of this blog.
I find myself unable to do any more work for a few hours, so I’ll post this as it is and extend the post later. I have created a very brief initial post on the Polymath blog. I suggest that mathematical comments should go there, whereas comments on the general Polymath issues discussed above would be more appropriate here. I’m hoping that people will regard the mathematical discussion as being in a preliminary stage, where the main purpose is to help as many people as possible get to the point where it is feasible for them to think about the problem, rather than to forge out ahead. However, it seems difficult (and not obviously right) to dictate this.