Generalized low solution of mathsf{RT}_k¹ problem

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 $I_3^1$.