Dilworth’s Decomposition Theorem for Posets1

The following theorem is due to Dilworth [12]: Let P be a partially ordered set. If the maximal number of elements in an independent subset (anti-chain) of P is k, then P is the union of k chains (cliques). In this article we formalize an elegant proof of the above theorem for finite posets by Perles [17]. The result is then used in proving the case of infinite posets following the original proof of Dilworth [12]. A dual of Dilworth’s theorem also holds: a poset with maximum clique m is a union of m independent sets. The proof of this dual fact is considerably easier; we follow the proof by Mirsky [16]. Mirsky states also a corollary that a poset of r × s + 1 elements possesses a clique of size r + 1 or an independent set of size s + 1, or both. This corollary is then used to prove the result of Erdős and Szekeres [13]. Instead of using posets, we drop reflexivity and state the facts about antisymmetric and transitive relations.