... ···<An.Inother words, the function γ restricted to the standard n2-variable submatrices of v is the lift of an r-coloring γ of the setF({1, ,L})n<. By the choice of L, there thusexist ... regularwords (that is, elements of WMk) 0-variable words, then the Hales-Jewett theoremcorresponds to the case n =0,t = 1 of Theorem 1.4.Theorem 1.4 Let k, r, n, t ∈ N be given. There exists M = M(k, ... We prove two extensions of the Hales-Jewett coloring theorem. The first is a polynomial version of a finitary case of Furstenberg and Katznelson’smultiparameter elaboration of a theorem, due to...