## Dilworth's Decomposition Theorem for Posets

The following theorem is due to Dilworth [8]: 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 [13]. The result is then used in proving the case of infinite posets following the original proof of Dilworth [8].

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 [11]. Mirsky states also a corollary that a poset of *r* x *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 [9].

Instead of using posets, we drop reflexivity and state the facts about anti-symmetric and transitive relations.

Published Online: 2010-07-08Published in Print: 2009-01-01Citation Information:Formalized Mathematics. Volume 17, Issue 4, Pages 223–232, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/v10037-009-0028-4, July 2010This content is open access.