The chain covering number of a poset with no infinite antichains

The chain covering number $\text{Cov}(P)$ of a poset $P$ is the least number of chains needed to cover $P$. For a cardinal $\nu$, we give a list of posets of cardinality and covering number $\nu$ such that for every poset $P$ with no infinite antichain, $\text{Cov}(P)\ge \nu$ 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 $\text{cf}(\nu )$ weakly compact. For $\nu ={\mathrm{\aleph }}_1$, a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal $\nu$.