On the indecomposability of $\omega^n$

TL;DR

This paper investigates the reverse mathematics of pigeonhole principles for finite powers of the ordinal ω, comparing four formulations and revealing connections to variants of Ramsey's Theorem for pairs.

Contribution

It introduces four natural formulations of pigeonhole principles for ω^n and analyzes their relative strengths, uncovering new links to Ramsey's Theorem variants.

Findings

Four formulations of pigeonhole principles for ω^n compared

Identifies two weak variants of Ramsey's Theorem for pairs

Provides insights into the reverse mathematics of ordinal powers

Abstract

We study the reverse mathematics of pigeonhole principles for finite powers of the ordinal $\omega$. Four natural formulations are presented and their relative strengths are compared. In the analysis of the pigeonhole principle for $\omega^2$, we uncover two weak variants of Ramsey's Theorem for pairs.