The chain covering number of a poset with no infinite antichains

The chain covering number Cov(P) of a poset P is the least number of chains needed to cover P. For an uncountable cardinal nu, we give a list of posets of cardinality and covering number nu such that for every poset P with no infinite antichain, Cov(P) >= nu if and only if P embeds a member of the list. This list has two elements if nu is a successor cardinal, namely [nu](2) and its dual, and four elements if nu is a limit cardinal with cf(nu) weakly compact. For nu = N-1, a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal nu.