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.
@article{bwmeta1.element.doi-10_2478_v10037-009-0028-4,
author = {Piotr Rudnicki},
title = {Dilworth's Decomposition Theorem for Posets},
journal = {Formalized Mathematics},
volume = {17},
year = {2009},
pages = {223-232},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10037-009-0028-4}
}
Piotr Rudnicki. Dilworth's Decomposition Theorem for Posets. Formalized Mathematics, Tome 17 (2009) pp. 223-232. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10037-009-0028-4/
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