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 GrahamRothschild 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 selfcontained proof" of anything. Polymath's DHJ proof does not use GR at all, on first glance.
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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 HalesJewett 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. 

