Dickson's lemma and weak Ramsey theory

TL;DR

This paper investigates the relationship between Dickson's lemma and weak Ramsey theory, establishing equivalences and exploring implications for various weak Ramsey principles within a foundational framework.

Contribution

It demonstrates the equivalence of a weak Paris--Harrington principle and miniaturized Dickson's lemma over RCA*_0, and analyzes consequences for weak Ramsey variants.

Findings

Equivalence between weak Paris--Harrington principle and miniaturized Dickson's lemma.

Implications for several variants of weak Ramsey's theorem.

Cascade of consequences for weak Ramsey principles.

Abstract

We explore the connections between Dickson's lemma and weak Ramsey theory. We show that a weak version of the Paris--Harrington principle for pairs in $c$ colors and miniaturized Dickson's lemma for $c$-tuples are equivalent over $\mathsf{RCA}_0^{\ast}$. Furthermore, we look at a cascade of consequences for several variants of weak Ramsey's theorem.