I am currently reading the recent preprint of Dodos, Kanellopoulos, Tyros, where the ambitiously short proof of Density Hales Jewett theorem is provided. The important ingredient is Graham-Rothschild theorem. The authors say that it follows from the HJ by some standard Ramsey arguments, but I can not find them myself, at least immediately. Is it written anywhere? Original paper of Graham and Rothschild looks too long for being used in "simple self-contained proof" of anything. Polymath's DHJ proof does not use GR at all, on first glance.
In Randall - McCutcheon's book "Elemental methods in ergodic Ramsey theory" a stronger version of GR's theorem is prooved about block subspaces (theorem 2.4.1). The only theorems you need in order to proove it is Hales-Jewett and Folkman's theorem (about finite unions which you can proove using Hales - Jewett theorem again).
Another proof is given in Graham - Rothschild -Spencer book about Ramsey theory using HJ and linear algebra.