COH, SRT 2 2 , and multiple functionals

We prove the following result: there is a family R = ⟨ 0 , 1 … ⟩ of subsets ω such that for every stable coloring c : [ ] 2 → k hyperarithmetical in and finite collection Turing functionals, an infinite homogeneous set H none finitely many functionals map ⊕ to cohesive R. This provides partial answer question computable combinatorics, whether COH omnisciently computably reducible SRT .