Reverse Mathematics and Ramsey Properties of Partial Orderings

Abstract

A partial ordering $\mathbb{P}$ is $n$-Ramsey if, for every coloring of $n$-element chains from $\mathbb{P}$ in finitely many colors, $\mathbb{P}$ has a homogeneous subordering isomorphic to $\mathbb{P}$. In their paper on Ramsey properties of the complete binary tree, Chubb, Hirst, and McNicholl ask about Ramsey properties of other partial orderings. They also ask whether there is some Ramsey property for pairs equivalent to ${\mathit{ACA}}_0$ over ${\mathit{RCA}}_0$.

A characterization theorem for finite-level partial orderings with Ramsey properties has been proven by the second author. We show, over ${\mathit{RCA}}_0$, that one direction of the equivalence given by this theorem is equivalent to ${\mathit{ACA}}_0$ (for $n\ge 3$), and the other is provable in ${\mathit{ATR}}_0$.

We answer Chubb, Hirst, and McNicholl’s second question by showing that there is a primitive recursive partial ordering $\mathbb{P}$ such that, over ${\mathit{RCA}}_0$, “$\mathbb{P}$ is 2-Ramsey” is equivalent to ${\mathit{ACA}}_0$.