Partitions of Countable Posets

For a countable bounded principal ideal poset \(P\) and a natural number \(r\), there exists a countable bounded principal ideal poset \(P’\) such that for an arbitrary \(r\)-colouring of the points (resp. two-chains) of \(P’\), a monochromatically embedded copy of \(P\) can be found in \(P’\). Moreover, a best possible upper bound for the height of \(P’\) in terms of \(r\) and the height of \(P\) is given.