The combinatorial equivalence of a computability theoretic question

TL;DR

This paper establishes an equivalence between a computability question related to colorings and a natural combinatorial problem, connecting computability theory with combinatorial principles like the Hales-Jewett theorem.

Contribution

It introduces a new equivalence between a computability problem and a combinatorial question, expanding understanding of their relationship.

Findings

Equivalence between a computability question and a combinatorial problem.

Generalization of the Hales-Jewett theorem for specific parameters.

Solved special cases of the combinatorial question.

Abstract

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.