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http://hdl.handle.net/11320/3548
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Pole DC | Wartość | Język |
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dc.contributor.author | Rudnicki, Piotr | - |
dc.date.accessioned | 2015-12-02T18:01:47Z | - |
dc.date.available | 2015-12-02T18:01:47Z | - |
dc.date.issued | 2009 | - |
dc.identifier.citation | Formalized Mathematics, Volume 17, Issue 4, 2009, Pages 223-232 | - |
dc.identifier.issn | 1426-2630 | - |
dc.identifier.issn | 1898-9934 | - |
dc.identifier.uri | http://hdl.handle.net/11320/3548 | - |
dc.description.abstract | 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. | - |
dc.language.iso | en | - |
dc.publisher | De Gruyter Open | - |
dc.title | Dilworth's Decomposition Theorem for Posets | - |
dc.type | Article | - |
dc.identifier.doi | 10.2478/v10037-009-0028-4 | - |
dc.description.Affiliation | University of Alberta, Edmonton, Canada | - |
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dc.description.references | Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990. | - |
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Występuje w kolekcji(ach): | Formalized Mathematics, 2009, Volume 17, Issue 4 |
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