Chain partitions of ordered sets

For natural numbers s and r and a finite poset P of height h, there exists a finite poset P of height H(s, r, h) such that for an arbitrary r-colouring of the s-chains of P, a monochromatically embedded copy of P can be found in P. A best possible upper bound for H(s, r, h) in terms of the well-known Ramsey numbers is given.