Ramsey’s theorem for singletons and strong computable reducibility

We answer a question posed by Hirschfeldt and Jockusch by showing that whenever k > ℓ Ramsey’s theorem for singletons and k-colorings, RT¹/_k, is not strongly computably reducible to the stable Ramsey’s theorem for _-colorings, SRT²/ ℓ Our proof actually establishes the following considerably stronger fact: given k > ℓ there is a coloring c: ω → ℓ such that for every stable coloring d: [ω]² → ℓ (computable from c or not), there is an infinite homogeneous set H for d that computes no infinite homogeneous set for c. This also answers a separate question of Dzhafarov, as it follows that the cohesive principle, COH, is not strongly computably reducible to the stable Ramsey’s theorem for all colorings, SRT²_<∞. The latter is the strongest partial result to date in the direction of giving a negative answer to the longstanding open question of whether COH is implied by the stable Ramsey’s theorem in ω-models of RCA₀.