  We show that a question of Miller and Solomon -- that whether there exists a
coloring $c:d^{<\omega}\rightarrow k$ that does not admit a $c$-computable
variable word infinite solution, is equivalent to a natural, nontrivial
combinatorial question. The combinatorial question asked whether there is an
infinite sequence of integers such that each of its initial segment satisfies a
Ramsian type property. This is the first computability theoretic question known
to be equivalent to a natural, nontrivial question that does not concern
complexity notions. It turns out that the negation of the combinatorial
question is a generalization of Hales-Jewett theorem. We solve some special
cases of the combinatorial question and obtain a generalization of Hales-Jewett
theorem on some particular parameters.
