I will give only a rather brief summary here, together with links to some comments that expand on what I say.
If we take our lead from known proofs of Roth’s theorem and the corners theorem, then we can discern several possible approaches (each one attempting to imitate one of these known proofs).
1. Szemerédi’s original proof.
Szemerédi proved a proof of a one-dimensional theorem. I do not know whether his proof has been modified to deal with the multidimensional case (this was not done by Szemerédi, but might perhaps be known to Terry, who recently digested and rewrote Szemerédi’s proof). So it is not clear whether one can base an entire argument for DHJ on this argument, but it could well contain ideas that would be useful for DHJ. For a sketch of Szemerédi’s proof, see this post of Terry’s, and in particular his discussion of question V. He also has links to comments that expand on what he says.
Ajtai and Szemerédi found a clever proof of the corners theorem that used Szemerédi’s theorem as a lemma. This gives us a direction to explore that we have hardly touched on. Very roughly, to prove the corners theorem you first use an averaging argument to find a dense diagonal (that is, subset of the form that contains many points of your set ). For any two such points and you know that the point does not lie in (if is corner free), which gives you some kind of non-quasirandomness. (The Cartesian semi-product arguments discussed in the 400s thread are very similar to this observation.) Indeed, it allows you to find a large Cartesian product in which is a bit too dense. In order to exploit this, Ajtai and Szemerédi used Szemerédi’s theorem to find long arithmetic progressions and of the same common difference such that has a density increment in . (I may have misremembered, but I think this must have been roughly what they did.) So we could think about what the analogue would be in the DHJ setting. Presumably it would be some kind of multidimensional Sperner statement of sufficient depth to imply Szemerédi’s theorem. A naive suggestion would be that in a dense subset of you can find a large-dimensional combinatorial subspace in which all the variable sets have the same size. If you apply this to a union of layers, then you find an arithmetic progression of layer-cardinalities. But this feels rather artificial, so here’s a question we could think about.
Question 1. Can anyone think what the right common generalization of Szemerédi’s theorem and Sperner’s theorem ought to be? (Sperner’s theorem itself would correspond to the case of progressions of length 2.)
The idea here is to prove the result by means of the following two steps.
1. If does not contain a combinatorial line, then it correlates with a set with some kind of atypical structure that we can describe.
2. Every set with that kind of structure can be partitioned into (or almost evenly covered by) a collection of combinatorial subspaces with dimension tending to infinity as tends to infinity.
One then finishes the argument as follows. If contains no combinatorial line, then find such that is a bit denser in than it is in . Cover with subspaces. By averaging we find that the density of is a bit too large in one of these subspaces. But now we are back where we started with a denser set so we can repeat. The iteration must eventually terminate (since we cannot have a density greater than 1) so if the initial was large enough then we must have had a combinatorial line.
One can also imagine splitting 1 up further as follows.
1a. If contains no combinatorial line, then contains too many of some other simple configuration.
1b. Any set that contains too many of those configurations must correlate with a structured set.
This is certainly how the argument goes in some of the proofs of related results.
This was the initial proposal, and we have not been concentrating on it recently, so I will simply refer the reader to the original post, and add the remark that if we manage to obtain a complete and global description of obstructions to uniformity, then regularity and triangle removal could be an alternative way of using this information to prove the theorem. And in the case of corners, this is a somewhat simpler thing to do.
For a proposal to base a proof on at least some of the ideas that come out of the proof of Furstenberg and Katznelson, see Terry’s comment 439, as well as the first few comments after it. Another helpful comment of Terry’s is his comment 460, which again is responded to by other comments. Terry has also begun an online reading seminar on the Furstenberg-Katznelson proof, using not just the original paper of Furstenberg and Katznelson but also a more recent paper of Randall McCutcheon that explains how to finish off the Furstenberg-Katznelson argument.
Suggestion. I have tried to do a bit of further summary in my first couple of comments. It would be good if others did the same, with the aim that anyone who reads this should be able to get a reasonable idea of what is going on without having to read too much of the earlier discussion.
Remark. There is now a wiki associated with this whole enterprise. It is in the very early stages of development but has some useful things on it already.