... · < An In other words, the function γ restricted to the standard n2 -variable submatrices of v is n the lift of an r-coloring γ of the set F ({1, , L}) < By the choice of L, there thus exist ... this fashion Proof of Theorem 1.7 Recall that our plan is to confine ourselves in the proof to the d = case The changes necessary to extend the proof to general d are minor and rather obvious, but ... words (that is, elements of Wk ) 0-variable words, then the Hales-Jewett theorem corresponds to the case n = 0, t = of Theorem 1.4 Theorem 1.4 Let k, r, n, t ∈ N be given There exists M = M (k,...