Ramsey Theory and Bounding in Arithmetic

TL;DR

This paper investigates the connections between Ramsey theorems, bounding schemes, and logical principles within a fragment of arithmetic, extending previous results and analyzing Weihrauch reducibility among these principles.

Contribution

It recasts and extends Hirst's results on Ramsey theory and bounding schemes, establishing equivalences and Weihrauch reductions among logical principles in a model of arithmetic.

Findings

Equivalence of BΣ₂, finite union of c.e. sets, and Infinite Pigeonhole Principle.

Weihrauch reducibility of Infinite Pigeonhole Principle to RT²₂.

Identification of principles equivalent to BΣ₃ and their reducibility to SRT²_{<∞}.

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: $B\Sigma_2$, 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 $B\Sigma_2$ which is Weihrauch reducible to $SRT^2_2$, see Theorem 5. We show that there is a principle which is equivalent with $B\Sigma_3$, see Theorem 6, and Weihrauch reducible to $SRT^2_{<\infty}$, Theorem 7. We discuss some equivalencies with $B\Sigma_{n-1}$, 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.