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 c...
