Ramsey Theory and Bounding in Arithmetic

Abstract

We explore the relation between various versions of Ramsey theorem and bounding schemes in model ${N}$ of a fragment of arithmetic $F$. Our goal is to recast, in a different framework, and extend some results of Hirst \cite{Hirst-1987}, see Theorem 1. We will extract Weihrauch reductions from Hirst's and similar proofs. Our results, informally stated in the our terminology, all inside ${N}$, follow: First the following are equivalent: $BΣ_2$, the finite union of finite c.e.\ sets is finite, and Infinite Pigeonhole Principle, see Theorem 3. We also discuss the Weihrauch relations between these logically equivalent principles, see Section 4. The Infinite Pigeonhole Principle is Weihrauch reducible to $RT^2_2$, see Theorem 4. There are also another principle logically equivalent to $BΣ_2$ which is Weihrauch reducible to $SRT^2_2$, see Theorem 5. We show that there is a principle which is equivalent with $BΣ_3$, see Theorem 6, and Weihrauch reducible to $SRT^2_{<\infty}$, Theorem 7. We discuss some equivalencies with $BΣ_{n-1}$, see Subsection 6.1, and then end with a problem Weihrauch reducible to $RT^{n+1}_{2}$, Subsection 6.2. The reader should be aware since we working within ${N}$ many of the standard definitions need to be adjusted to work. Due to the expository nature of this short paper, these definitions are sprinkled throughout the paper. A quick read of the paper from start to finish will provide a better understanding of the ideas involved rather than a careful reading of theorems.