## DHJ write-up and other matters

This short post is in response to Jozsef Solymosi’s request for a new DHJ thread, since the previous one has become rather long and unwieldy. We’ve stopped numbering comments now, and the main purpose of the post is so that people can continue the discussion of the write-up of the proof of DHJ(k). Thanks mostly to the efforts of Ryan O’Donnell, we now have a complete draft. See also this write-up of DHJ(3) by Jozsef.

While I’m writing, I thought I’d take the opportunity to say that I am not intending to post much over the next two or three months, either here on on the Tricki. That’s because I have three more or less completed research projects that need to be properly finished (one of which is DHJ) and I owe it to my coauthors to get them done. So the plan is to clear my backlog over the summer and then come back, refreshed and ready to go, in the autumn. At that point I plan several Tricki articles (more advanced than most of the ones I’ve written so far). I also plan to start a new polymath project. Or rather, I have a file in which I have written plans for ten polymath projects, so what I’ll probably do is explain briefly what they are and try to get some idea of what appeals to people most. I am excited about several of these possible projects, so whatever we do I will be disappointed about the ones we don’t do. I may well have an online vote about it, but first I have to decide what the results of the vote will be.

### 11 Responses to “DHJ write-up and other matters”

1. Matemáticos colaboram num projecto aberto de investigação conjunta através do blogue do Professor Timothy Gowers « problemas | teoremas Says:

[…] Em post de hoje (DHJ write-up and other matters) o Professor Gowers anuncia […]

2. Ryan O'Donnell Says:

Here are the write-up repositories on the wiki for the two papers.

3. Ryan O'Donnell Says:

I was looking at Shkredov’s paper today, http://arxiv.org/PS_cache/math/pdf/0405/0405406v1.pdf

We really must be sure to cite it, as our double-inductive proof structure is very much the same as his. The “show a relative density increment on a somewhat-structured set” part is Proposition 4.10 in his paper, and the “inductively partition somewhat-structured sets into disjoint very-structured sets (plus a tiny error set)” part is Theorem 5.2 in his paper.

4. Ryan O'Donnell Says:

Also, just so it doesn’t get forgotten, Kevin O’Bryant made the good point on the earlier thread that the paper should make sure to state the best known lower bounds for DHJ(k) (proved in the other paper).

5. D. Eppstein Says:

So the plan is to clear my backlog over the summer and then come back, refreshed and ready to go, in the autumn.

I hope you’re better at actually carrying out your plans for what you want to get accomplished over the summers than I am.

6. Ryan O'Donnell Says:

I finally took the time to, you know, actually read the Gunderson-Rodl-Sidirenko paper (and the Erdos paper it relies on). I believe one can condense the argument and get marginally better bounds (not good enough to improve the DHJ bounds). Here is the pdf and tex; we could possibly consider including it in the paper.

7. OU Math Club Says:

[…] are currently working on writing up the results of that project as a research paper for publication.  The author of the […]

8. Kevin O'Bryant Says:

At the bottom of page two, we mention that the multi-dimensional theorem follows from the lines version. According to Math Sci-Net, this is the content of:

MR0734978 (85d:05068)
Brown, T. C.(3-SFR); Buhler, J. P.(1-REED)
Lines imply spaces in density Ramsey theory.
J. Combin. Theory Ser. A 36 (1984), no. 2, 214–220.
05B25 (51E30)

Let a finite field $F$ with $q$ elements and $\varepsilon>0$ be fixed. The affine line conjecture is the assertion that for $n$ sufficiently large if $V$ is a vector space of dimension $n$ over $F$ and $A\subset V$ with $\vert A\vert \ge\varepsilon\vert V\vert$ then $A$ contains an affine line. This roughly is the density version of the Hales-Jewett theorem and is a leading open question in Ramsey theory today. Let the affine space conjecture denote the assertion that given $F$, $\varepsilon>0$, and a positive integer $k$ then, for $n$ sufficiently large with $A$ as above, $A$ contains an affine $k$-space. The authors show that the affine line conjecture implies the affine space conjecture.

Reviewed by J. Spencer

9. Ryan O'Donnell Says:

A new version has been posted to the wiki.

10. Christian Elsholtz Says:

I just looked at latest version of the DHJ paper, and noticed that the reference given to Behrend’s construction, at the bottom of page 2,
links to his 1938 paper in Casopis pro pestovani….
I doubt that the 1938 paper contains what one usually means by Behrend’s construction. A link to that paper is here
http://dml.cz/dmlcz/122006

Behrend’s construction in the usual sense is in his 1946 paper in the Proceedings of the National Academy of Sciences, a link is here:
http://www.pnas.org/content/32/12/331.citation