After a hiatus of several months, I’ve made an effort to advance the writing of the second Polymath1 paper, entitled “Density Hales-Jewett and Moser numbers“.  This is in part due to a request from the Szemeredi 60th 70th birthday conference proceedings (which solicited the paper) to move the submission date up from April to February.  (Also, the recent launch of Polymath5 on Tim Gowers blog reminds me that I should get this older project out of the way.)

The current draft of the paper is here, with source files here.  I have been trimming the paper, in particular replacing some of the auxiliary or incomplete material in the paper with references to pages on the polymath wiki instead.  Nevertheless this is still a large paper, at 51 pages.  It is now focused primarily on the computation of the Density Hales-Jewett numbers c_{n,3} and the Moser numbers c'_{n,3} for all n up to 6, with the latter requiring a significant amount of computer assistance.

There are a number of minor issues remaining with the paper:

  1. A picture of a Fujimura set for the introduction would be nice.
  2. In the proof of Theorem 1.3 (asymptotic lower bound for DHJ numbers), it is asserted without proof that the circulant matrix with first row 1,2,…,k-1 is nonsingular.  One can prove this claim by computing the Fourier coefficients \sum_{j=1}^{k-1} j e^{2\pi i j t / (k-1)} for all t, but is there a slicker way to see this (e.g. by citing a reference?).
  3. Reference [15] (which is Komlos’s lower bound on the Moser numbers) is missing a volume number.  The reference is currently given as
    J. Komlos, solution to problem P.170 by Leo Moser, Canad. Math.. Bull. vol  ???  (1972), 312-313, 1970.

Finally, the text probably needs to be proofread one or two more times before it is ready to go, hopefully by early February.  There is still also one last opportunity to propose non-trivial restructuring of the paper (in particular, if there are other ways to trim the size of the paper, this may be worth looking into).