Cross-constraint basis theorems and products of partitions

TL;DR

This paper surveys and extends a technique to prove separation theorems between products of Ramsey-type theorems, demonstrating non-reducibility between certain variants over computable reducibility.

Contribution

It introduces an extension of Lu Liu's technique to establish new separation results for products of Ramsey theorems in computability theory.

Findings

Ramsey's theorem for n-tuples and three colors is not computably reducible to finite products of the two-color version.

The paper extends existing techniques to prove separation theorems in the context of computable reducibility.

Provides a unified framework for analyzing relationships between different Ramsey-type theorems.

Abstract

We both survey and extend a new technique from Lu Liu to prove separation theorems between products of Ramsey-type theorems over computable reducibility. We use this technique to show that Ramsey's theorem for $n$-tuples and three colors is not computably reducible to finite products of Ramsey's theorem for $n$-tuples and two colors.