Cohesive Avoidance and Strong Reductions

An open question in reverse mathematics is whether the cohesive principle, COH, is implied by the stable form of Ramsey’s theorem for pairs, SRT2, in ω-models of RCA0. One typical way of establishing this implication would be to show that for every sequence R⃗ of subsets of ω, there is a set A that is ∆2 in R⃗ such that every infinite subset of A or A computes an R⃗-cohesive set. In this article, this is shown to be false, even under far less stringent assumptions: for all natural numbers n ≥ 2 and m < 2n, there is a sequence R⃗ = ⟨R0, . . . , Rn−1⟩ of subsets of ω such that for any partition A0, . . . , Am−1 of ω hyperarithmetical in R⃗, there is an infinite subset of some Aj that computes no set cohesive for R⃗. This complements a number of previous results in computability theory on the computational feebleness of infinite sets of numbers with prescribed combinatorial properties. The proof is a forcing argument using an adaptation of the method of Seetapun showing that every finite coloring of pairs of integers has an infinite homogeneous set not computing a given non-computable set.