Ramsey properties of countably infinite partial orderings (Q1953437)

Summary: 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.