RAMSEY-LIKE THEOREMS AND MODULI OF COMPUTATION

Ramsey's theorem asserts that every k-coloring of [omega](n) admits an infinite monochromatic set. Whenever n >= 3, there exists a computable k-coloring of [omega](n) whose solutions compute the halting set. On the other hand, for every computable k-coloring of [omega](2) and every noncomputable set C, there is an infinite monochromatic set H such that C not less than or equal to(T) H. The latter property is known as cone avoidance. In this article, we design a natural class of Ramsey-like theorems encompassing many statements studied in reverse mathematics. We prove that this class admits a maximal statement satisfying cone avoidance and use it as a criterion to re-obtain many existing proofs of cone avoidance. This maximal statement asserts the existence, for every k-coloring of [omega](n), of an infinite subdomain H subset of omega over which the coloring depends only on the sparsity of its elements. This confirms the intuition that Ramsey-like theorems compute Turing degrees only through the sparsity of its solutions.