Dickson's lemma and weak Ramsey theory

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 ${\mathsf{R}\mathsf{C}\mathsf{A}}_0^{\ast }$. Furthermore, we look at a cascade of consequences for several variants of weak Ramsey’s theorem.