# COH, SRT22, and multiple functionals

## TL;DR

This paper constructs a specific family of sets to demonstrate limitations in the computability of homogeneous sets for stable colorings, advancing understanding of the relationship between the principles RT^2_2 and OH in reverse mathematics.

## Contribution

It provides a new partial result showing that certain stable colorings do not allow for the computation of infinite cohesive sets via finitely many Turing functionals, addressing open problems in reverse mathematics.

## Key findings

- Demonstrates existence of a set family with specific computability properties
- Shows limitations in reducing OH to RT^2_2 using computable functionals
- Advances understanding of the RT^2_2 vs. OH problem

## Abstract

We prove the following result: there is a family $R = \langle R_0,R_1,\ldots \rangle$ 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 $\mathsf{SRT}^2_2$.