Without anyone being particularly aware of it, a race has been taking place. Which would happen first: the comment count reaching 1000, or the discovery of a new proof of the density Hales-Jewett theorem for lines of length 3? Very satisfyingly, it appears that DHJ(3) has won. If this were a conventional way of producing mathematics, then it would be premature to make such an announcement — one would wait until the proof was completely written up with every single i dotted and every t crossed — but this is blog maths and we’re free to make up conventions as we go along. So I hereby state that I am basically sure that the problem is solved (though not in the way originally envisaged).
Why do I feel so confident that what we have now is right, especially given that another attempt that seemed quite convincing ended up collapsing? Partly because it’s got what you want from a correct proof: not just some calculations that magically manage not to go wrong, but higher-level explanations backed up by fairly easy calculations, a new understanding of other situations where closely analogous arguments definitely work, and so on. And it seems that all the participants share the feeling that the argument is “robust” in the right way. And another pretty persuasive piece of evidence is that Tim Austin has used some of the ideas to produce a new and simpler proof of the recurrence result of Furstenberg and Katznelson from which they deduced DHJ. His preprint is available on the arXiv.
Better still, it looks very much as though the argument here will generalize straightforwardly to give the full density Hales-Jewett theorem. We are actively working on this and I expect it to be done within a week or so. (Work in progress can be found on the polymath1 wiki.) Better even than that, it seems that the resulting proof will be the simplest known proof of Szemerédi’s theorem. (There is one other proof, via hypergraphs, that could be another candidate for that, but it’s slightly less elementary.)
I have lots of thoughts about the project as a whole, but I want to save those for a different and less mathematical post. This one is intended to be the continuation of the discussion of DHJ(3), and now DHJ(k), into the 1000s. More precisely, it is for comments 1000-1049.