$\mathsf{RT}_2^2$ does not imply $\mathsf{WKL}_0$

TL;DR

The paper demonstrates that the combinatorial principle $ ext{RT}_2^2$ does not imply the subsystem $ ext{WKL}_0$ in reverse mathematics by constructing specific sets with particular degree properties.

Contribution

It provides a new separation result in reverse mathematics showing $ ext{RT}_2^2$ does not imply $ ext{WKL}_0$ using degree-theoretic constructions.

Findings

$ ext{RT}_2^2$ does not imply $ ext{WKL}_0$ in $ ext{RCA}_0$.

Existence of infinite subsets with non-PA degrees.

Construction of sets avoiding PA-degree when combined with a non-PA-degree set.

Abstract

We prove that $\mathsf{RCA}_0+\mathsf{RT}_2^2\not\rightarrow \mathsf{WKL}_0$ by showing that for any set $C$ not of PA-degree and any set $A$, there exists an infinite subset $G$ of $A$ or $\bar{A}$, such that $G\oplus C$ is also not of PA-degree.