Dilworth’s theorem

Theorem.

If $P$ is a poset with width , then $w$ is also the smallest integer such that $P$ can be written as the union of $w$ chains.

Remark. The smallest cardinal $c$ such that $P$ can be written as the union of $c$ chains is called the chain covering number of $P$. So Dilworth’s theorem says that if the width of $P$ is finite, then it is equal to the chain covering number of $P$. If $w$ is infinite, then statement is not true. The proof of Dilworth’s theorem and its counterexample in the infinite case can be found in the reference below.