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

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 $0^{\prime }$-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 $I_3^1$.

14 pages