Ramsey Properties of Countably Infinite Partial Orderings

A partial ordering ℙ is chain-Ramsey if, for every natural number n and every coloring of the n-element chains from ℙ in finitely many colors, there is a monochromatic subordering ℚ isomorphic to 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