Ramsey Properties of Countably Infinite Partial Orderings

Abstract

A partial ordering $\mathbb P$ is chain-Ramsey if, for every natural number $n$ and every coloring of the $n$-element chains from $\mathbb P$ in finitely many colors, there is a monochromatic subordering $\mathbb Q$ isomorphic to $\mathbb P$.  Chain-Ramsey partial orderings stratify naturally into levels.  We show that a countably infinite partial ordering with finite levels is chain-Ramsey if and only if it is biembeddable with one of a canonical collection of examples constructed from certain edge-Ramsey families of finite bipartite graphs.  A similar analysis applies to a large class of countably infinite partial orderings with infinite levels.