Polymath's combinatorial proof of the density Hales-Jewett theorem,
arXiv:1205.7084
, May 2012. Comment (December 2012). Much shorter (cca 10 pages)
combinatorial proof of the density Hales-Jewett theorem, than
Polymath's, was given by P. Dodos, V. Kanellopoulos and K. Tyros in
arXiv:1209.4986
in September 2012. This seems to me at present to be the simplest
(self-contained) proof of the DHJ theorem and, a fortiori, of
Szemerédi's theorem.