The chain covering number $\Cov(P)$ of a poset $P$ is the least number of chains needed to cover $P$. For a cardinal $ν$, we give a list of posets of cardinality and covering number $ν$ such that for every poset $P$ with no infinite antichain, $\Cov(P)\geq ν$ 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 $ν= \aleph_1$, a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal $ν$.