| Summary: | The chain covering number $\operatorname{Cov}(P)$ 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, $\operatorname{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 $\operatorname{cf}(\nu )$ 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 $.
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