  The chain covering number $\Cov(P)$ 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, $\Cov(P)\geq \nu$ 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 $\nu= \aleph_1$, a list was given by the first
author; his construction was extended by F. Dorais to every infinite successor
cardinal $\nu$.
