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 ν, we give a list of posets of cardinality and covering number ν such that for every poset P with no infinite antichain, Cov(P) ≥ ν if and only if P embeds a member of the list. This list has two elements if ν is a successor cardinal, namely [ν]² and its dual, and four elements if ν is a limit cardinal with cf(ν) weakly compact. For ν = ℵ₁, a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal ν.