Generalized low solution of $\mathsf{RT}_k^1$ problem

TL;DR

This paper investigates the computational complexity of solutions to instances of the Ramsey theorem for singletons, proving the existence of non-trivial generalized low solutions for all such instances and addressing open questions in the field.

Contribution

It demonstrates that every RT_k^1-instance has a non-trivial generalized low solution and constructs a specific RT_3^1-instance with unique computational properties.

Findings

Every RT_k^1-instance admits a non-trivial generalized low solution.

Constructed a RT_3^1-instance with solutions that do not compute solutions of certain RT_2^1-instances.

Abstract

We study the "coding power" of an arbitrary $\mathsf{RT}_k^1$-instance. We prove that every $\mathsf{RT}_k^1$-instance admit non trivial generalized low solution. This is somewhat related to a problem proposed by Patey. We also answer a question proposed by Liu, i.e., we prove that there exists a $\mathbf{0}'$-computable $\mathsf{RT}_3^1$-instance, $I_3^1$, such that every $\mathsf{RT}_2^1$-instance admit a non trivial solution that does not compute any non trivial solution of $I_3^1$.