I’ve reached the first point where it isn’t almost trivial to generalize the argument we have for DHJ(3) to an argument for DHJ(k). I think it’s going to be a sort of middling difficulty — that is, you can’t do it in five minutes but you know you’ll get there in at most a few hours. (Actually, I think it may be easier than that, but I haven’t yet tried.) Over on the wiki, I have written up part of an argument that generalizes to the fact that a dense line-free set (in equal-slices measure) correlates with a dense 12-set.

The problem in the case is to prove that the lower-complexity set you get is dense. The proof for uses a quantitative Sperner theorem (namely that you get a positive proportion of all possible combinatorial lines if you start with a dense set — all with respect to equal-slices measure). The proof for needs a similar statement about combinatorial lines in It should be obtainable from DHJ(k-1) by a suitable averaging argument, but it’s not one I can do in my head.

]]>I will probably focus the (limited) time I have available for this thread on trying to explicate Austin’s proof in finitary language. I know you guys don’t actually have access to it yet, but let me try to informally describe some of the details. Take all this *cum grano solis*; I have not yet fully digested the argument and some of the details may be slightly or perhaps even massively incorrect.

Let me take [3] = {0,1,2} (rather than {1,2,3}) for sake of arbitrarily fixing the conventions (we’ve been a bit inconsistent on this). Let’s define a trilinear form on functions by the formula

where varies along combinatorial lines with respect to some measure that I will intentionally leave vague. DHJ(3) is equivalent to the following “triangle-removalish” type statement:

**DHJ(3)’**: Let be such that . Then .

Roughly speaking, Austin’s strategy is to “regularise” the situation so that 12-sets, 01-sets are “relatively independent” over a common algebra of 1-sets, and similarly for the 12-sets and 02-sets, etc., with the 0-sets, 1-sets, 2-sets themselves being relatively independent over -sets (which, for us, I think means “unions of large subspaces”, and which can be ignored by passing to a large subspace). I don’t understand this part well yet, but it is analogous to the Szemeredi regularity lemma. There is also a preliminary reduction to “strong stationarity” which means, roughly, that the statistics of various 01-sets, etc. (e.g. the density of an overlap between a relevant 01-set and a relevant 02-set) doesn’t change of we freeze a bounded number of coordinates. This reduction is obtained via Graham-Rothschild and is going to be hideously expensive as regards quantitative bounds, but never mind that for now.

1. Once one has this regularisation, this makes DHJ(3)’ is easy when f is an (indicator of a) (basic) 12-set, g is a basic 02-set, and h is a basic 01-set, much as it is easy to find corners connecting three sets A, B, C in when those sets are Cartesian products in the right fashion. Indeed things seem to collapse to DHJ(2.5) in this case. (This is analogous to the triangle removal lemma for three bipartite graphs when each of the three graphs is a complete bipartite graphs between one cell in each vertex set.)

2. Next, this implies DHJ(3)’ when f,g,h are non-basic 12-sets, 02-sets, and 01-sets, i.e. finite unions of basic sets (or more precisely finite linear combinations of indicators of basic sets), but where the lower bound depends on the complexity of the partition into basic sets. (This is analogous to the triangle removal lemma for unions of complete bipartite graphs between cells.)

3. Next, this implies DHJ(3)’ when f (and similarly g, h) are “Borel 12-sets” (to continue the topological analogy much as Borel sets can be approximated by open sets), which means that given any , one can approximate f to within by a non-basic 12-set of bounded complexity. This is because pigeonhole ensures that there are a lot of non-basic 12-sets which are 99% occupied by f, and one will be able to get this case from applying Case 2 to these “rich” sets, and using some relative independence properties. (This is analogous to triangle removal for unions of 99%-complete bipartite graphs between cells.)

4. Next, we obtain DHJ(3)’ when f (and similarly g,h) is the sum of a Borel 12-set and something which is highly orthogonal to all basic 12-sets, including the very small basic 12-sets coming from the cells of an approximation to the Borel 12-set. This is basically because the guy which is highly orthogonal to all basic 12-sets is so uniform as to have essentially no contribution to . (This is analogous to triangle removal for a triplet of bipartite graphs which have been regularised.)

5. Finally, we have a regularity lemma that tells us that arbitrary f,g,h decompose in this fashion (possibly after localising to a large subspace). This is a “soft” energy increment argument, analogous to that in the regularity lemma. One has to keep freezing coordinates while performing this increment argument, so it is important that one has the strong stationarity property first before one sets up the regularity argument.

Maybe I’ll try to write a more coherent version of the above on the wiki at some point.

]]>Randall, there may be room for disagreement over your slogan, but I can’t help liking it …

]]>The hypergraph regularity/removal proof of (multidimensional) Szemeredi is not too bad, actually, despite its reputation. The ergodic version of it, which Tim Austin wrote up in the arXiv a few months back, is perhaps a touch simpler than the Furstenberg-Katznelson proof based on repeated extensions by relatively almost periodic functions, being instead based on extending the entire system up to a more “pleasant” system enjoying a number of useful relative independence properties.

]]>Regarding what Tim (Gowers) said about an easy proof of Szemeredi materializing, as well as what Terry said about avoiding relative almost periodicity (which seems to be exactly what makes this proof easy as well), it seems natural whether one of us should think about writing up carefully a “book proof of Szemeredi’s theorem”. For starters, it seems to me that this might entail proving Jozsef’s comment no. 2 from a multi-dim Sperner type of hypothesis, then pushing an induction to k-dimensional corners ala Tim’s 886. I am hoping it could be easier in the details than DHJk; that should be clear by the end of the first step, though.

]]>Having been following progress here (albeit only in fits and starts) and talking with Terry about the ideas that have come out, it struck me late last week that in the infinitary world of stochastic processes that Furstenberg and Katznelson move to, the approach raised here for obtaining obstructions to uniformity that are built from ij-sets can actually be coupled to a lot of machinery that’s already known from other things to give a new infinitary proof of their multiple recurrence result, without anything else being required. In particular, it uses an infinitary analog of `energy increment’ to improve the structure of a stochastic process, and then an appeal to an `infinitary hypergraph removal lemma’ originally motived by some work of Terry on infinite random hypergraphs, both which I recently used to play a similar game around the multidimensional Szemeredi Theorem (arXiv 0808.2267, in case it’s of interest).

In fact, it turned out that this could be written up completely in just a couple of days by judiciously cutting, pasting and re-notating writeups of other things, so this is now done and on the arXiv: once it becomes publicly visible it’ll be at 0903.1633. I feel I should possibly offer my assurances that I wouldn’t have rushed from a moment of realization to completing a preprint if it really hadn’t been so very quick and mechanical from that point on, without requiring any input of new ideas from me.

For what it’s worth, I’ve thought only briefly about finitizing this approach and Terry has already said most of what I could say. As it stands it will require a preliminary heavy appeal to Graham-Rothschild (Carlson-Simpson, in the infinitary world) and then proceeding in close analogy with hypergraph removal strategies. So it is rather removed from the density-increment approach that I think is now mainly being pursued here, and would look set to give much worse bounds unless some other new idea can remove the reliance on Graham-Rothschild.

]]>I’ve been busy, so I haven’t been able to stop by much recently, but things look pretty good at this point. I agree that the DHJ(3) Ajtai-Szemeredi sketch looks pretty solid. (An amusing side note: when I had Ajtai-Szemeredi described to me, I thought that they were already doing what we were doing now, i.e. getting correlation with an unstructured Cartesian product and then partitioning that product into grids. So I was a little confused when Tim was insisting that what we were doing was not quite Ajtai-Szemeredi… but now I see the subtle difference between the two approaches.)

It looks like Tim Austin has come up with an alternate proof that is also based on correlation with 12-sets, etc. but is based on triangle-removal type strategies rather than density-increment strategies. It also requires a preliminary use of Graham-Rothschild to regularise a large number of statistics so as to make them stable under freezing of coordinates, and so is likely to give poorer bounds. But it is closer in spirit to the original intent of Polymath1. I believe Tim will come on here himself to report on this soon (he’s working in an ergodic theory setting), and I will focus on trying to finitise it. (It’s likely to be cleaner than the finitisation of Furstenberg-Katznelson, because one does not have to deal with relative almost periodicity.)

]]>Meta comment: Something happened with the wiki. It seems that it has been hacked. Be careful with the links there.

]]>I finished the 1% fleshing out required in the article proving that line-free sets correlate with 12-sets, including all the passing back and forth between uniform and equal-slices measures. I think the only bit remaining undone here is instantiating all the parameters.

]]>My main teaching days are Mondays and Tuesdays this term, and today and tomorrow are the last two such days of term. So I’ll be fairly busy, but I hope I’ll still have a bit of time for blogging and wikification. Here I want, as a pre-wikification exercise, to sketch a proof that DHJ(k) implies multidimensional DHJ(k). I’ve woken up with the feeling that DHJ(k) is going to go through almost as easily as DHJ(3). If that is the case, it will be unexpected for two reasons. First, it will give a proof of Szemerédi’s theorem that has a strong claim to be the simplest known. (The only rival I can think of is a particularly clean approach via infinitary hypergraphs, due to Elek and Szegedy, but I may be wrong.) Secondly, it would be the first proof of Szemerédi’s theorem for which “the general case is the case k=3”. By that I mean that in all other proofs you have to go at least as far as k=4 before it’s obvious how to generalize, and in some you have to go to k=5. (Perhaps a true understanding of the problem would require a proof that generalizes straightforwardly from the k=2 case …)

Back to multidimensional DHJ(k). Here’s what I think works. Let be a density- subset of and let be large enough so that every subset of of density at least contains a combinatorial line. Now split up into For a proportion at least of the points y in the set of such that has density at least . Therefore, by DHJ(k) (with ) we have a combinatorial line. Since there are fewer than to choose from, by the pigeonhole principle we can find a combinatorial line and a set of density in such that whenever and And now by induction we can find an -dimensional subspace in and we’re done.

This gives a truly horrible bound, and should mean that if DHJ(k) goes through as I expect (and Randall also expects, I’m glad to see from 886.2), the bound that comes out at the end will probably be of Ackermann type, so it will be comparable to the bounds that come out of the hypergraph approach. (A small challenge that I know some people out there would enjoy is to try to see how this approach to Szemerédi fits in with the general philosophy that all the different proofs are at some deep level manifestations of closely related ideas. There are distinct echoes of hypergraphs in this proof, and yet it is far easier than hypergraph regularity and counting — what is going on? Possibly that we are “cheating” by continually passing to subspaces, but why can’t we do that with hypergraphs? Or can we? Perhaps there’s a way of passing to subgraphs without throwing away too many degenerate simplices. Hmm … I quite like that but no time to pursue it just at the moment.)

]]>Okay, using the above mentality I was able to rewrite in my own words Tim’s proof that line-free sets correlate with 12-sets. I added these words to the wiki, modulo the passing from uniform density to equal-slices density (which is still in that article and also partially here). It’s pretty late at night for me, so I hope I got it right.

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