Reverse mathematics of the finite downwards closed subsets of $\mathbb{N}^k$ ordered by inclusion and adjacent Ramsey for fixed dimension

TL;DR

This paper establishes the equivalence between the well-partial orderedness of finite downwards closed subsets of ^k and the well-foundedness of a specific ordinal, also analyzing the reverse mathematical strength of Friedman's adjacent Ramsey theorem.

Contribution

It proves the conjecture linking well-partial orderedness to ordinal well-foundedness and explores the reverse mathematical strength of Friedman's adjacent Ramsey theorem.

Findings

Equivalence between well-partial orderedness and ordinal ^{^}

Reverse mathematical analysis of Friedman's adjacent Ramsey theorem

Clarification of the logical strength of these combinatorial principles

Abstract

We show that the well-partial orderedness of the finite downwards closed subsets of $\mathbb{N}^k$ ,ordered by inclusion, is equivalent to the well-foundedness of the ordinal $\omega^{\omega^\omega}$. This was conjectured to be the case by Hatzikiriakou and Simpson. Since we use Friedman's adjacent Ramsey theorem for fixed dimensions in the upper bound, we also give a treatment of the reverse mathematical status of that theorem.