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The polymath1 project has just uploaded to the arXiv the paper “A new proof of the density Hales-Jewett theorem“, to be submitted shortly.  Special thanks here go to Ryan O’Donnell for performing the lion’s share of the writing up of the results, and to Tim Gowers for running a highly successful online mathematical experiment.

I’ll state the main result in the first non-trivial case $k=3$ for simplicity, though the methods extend surprisingly easily to higher $k$ (but with significantly worse bounds).  Let $c_{n,3}$ be the size of the largest subset of the cube $[3]^n = \{1,2,3\}^n$ that does not contain any combinatorial line.    The density Hales-Jewett theorem of Furstenberg and Katznelson shows that $c_{n,3} = o(3^n)$.  In the course of the Polymath1 project, the explicit values

$c_{0,3} = 1; c_{1,3} = 2; c_{2,3} = 6; c_{3,3} = 18; c_{4,3} = 52; c_{5,3} =150; c_{6,3} = 450$

were established, as well as the asymptotic lower bound

$c_{n,3} \geq 3^n \exp( - O( \sqrt{ \log n } ) )$

(actually we have a slightly more precise bound than this).  The main result of this paper is then

Theorem. ($k=3$ version) $c_{n,3} \ll 3^n / \log^{1/2}_* n$.

Here $\log_* n$ is the inverse tower exponential function; it is the number of times one has to take (natural) logarithms until one drops below 1.  So it does go to infinity, but extremely slowly.  Nevertheless, this is the first explicitly quantitative version of the density Hales-Jewett theorem.

The argument is based on the density increment argument as pioneered by Roth, and also used in later papers of Ajtai-Szemerédi and Shkredov on the corners problem, which was also influential in our current work (though, perhaps paradoxically, the generality of our setting makes our argument simpler than the above arguments, in particular allowing one to avoid use of the Fourier transform, regularity lemma, or Szemerédi’s theorem).   I discuss the argument in the first part of this previous blog post.

I’ll end this post with an open problem.  In our paper, we cite the work of P. L. Varnavides, who was the first to observe the elementary averaging argument that showed that Roth’s theorem (which showed that dense sets of integers contained at least one progression of length three) could be amplified (to show that there was in some sense a “dense” set of arithmetic progressions of length three).  However, despite much effort, we were not able to expand “P.” into the first name.  As one final task of the Polymath1 project, perhaps some readers with skills in detective work could lend a hand in finding out what Varnavides’ first name was? Update, Oct 22: Mystery now solved; see comments.

As part of the polymath1 project, I would like to set up a reading seminar on this blog for the following three papers and notes:

1. H. Furstenberg, Y. Katznelson, “A density version of the Hales-Jewett theorem for k=3“, Graph Theory and Combinatorics (Cambridge, 1988). Discrete Math. 75 (1989), no. 1-3, 227–241.
2. R. McCutcheon, “The conclusion of the proof of the density Hales-Jewett theorem for k=3“, unpublished.
3. H. Furstenberg, Y. Katznelson, “A density version of the Hales-Jewett theorem“, J. Anal. Math. 57 (1991), 64–119.

As I understand it, paper #1 begins the proof of DHJ(3) (the k=3 version of density Hales-Jewett), but the proof is not quite complete, and the notes in #2 completes the proof using ideas from both paper #1 and paper #3.  Paper #3, of course, does DHJ(k) for all k.  For the purposes of the polymath1 project, though, I think it would be best if we focus exclusively on k=3.

While this seminar is of course related in content to the main discussion threads in the polymath1 project, I envision this to be a more sedate affair, in which we go slowly through various sections of various papers, asking questions of each other along the way, and presenting various bits and pieces of the proof.  The papers require a certain technical background in ergodic theory in order to understand, but my hope is that if enough other people (in particular, combinatorialists) ask questions here (and “naive” or “silly” questions are strongly encouraged) then we should be able to make a fair amount of the arguments here accessible.  I also hope that some ergodic theorists who have been intending to read these papers already, but didn’t get around to it, will join with reading the papers with me.

This is the first time I am trying something like this, and so we shall be using the carefully thought out protocol known as “making things up as we go along”.  My initial plan is to start understanding the “big picture” (in particular, to outline the general strategy of proof), while also slowly going through the key stages of that proof in something resembling a linear order.  But I imagine that the focus may change as the seminar progresses.

I’ll start the ball rolling with some initial impressions of paper #1 in the comments below.  As with other threads in this project, I would like all comments to come with a number and title, starting with 600 and then incrementing (the numbers 1-599 being reserved by other threads in this project).