COH, SRT22, and multiple functionals

Abstract

We prove the following result: there is a family $R=⟨R_0,R_1,\dots ⟩$ of subsets of $\omega$ such that for every stable coloring $c:[\omega {]}^2\to k$ hyperarithmetical in $R$ and every finite collection of Turing functionals, there is an infinite homogeneous set $H$ for $c$ such that none of the finitely many functionals map $R\oplus H$ to an infinite cohesive set for $R$. This extends the current best partial results towards the ${\mathsf{SRT}}_2^2$ vs. $\mathsf{COH}$ problem in reverse mathematics, and is also a partial result towards the resolution of several related problems, such as whether $\mathsf{COH}$ is omnisciently computably reducible to ${\mathsf{SRT}}_2^2$.

13 pages