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• # Artykuł - szczegóły

## Formalized Mathematics

2009 | 17 | 4 | 223-232

## Dilworth's Decomposition Theorem for Posets

EN

### Abstrakty

EN
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.

223-232

wydano
2009-01-01
online
2010-07-08

### Twórcy

autor
• University of Alberta, Edmonton, Canada

### Bibliografia

• [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
• [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
• [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
• [4] Grzegorz Bancerek. Bounds in posets and relational substructures. Formalized Mathematics, 6(1):81-91, 1997.
• [5] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93-107, 1997.
• [6] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• [7] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
• [8] R. P. Dilworth. A Decomposition Theorem for Partially Ordered Sets. Annals of Mathematics, 51(1):161-166, 1950.[Crossref]
• [9] P. Erdős and G. Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463-470, 1935.
• [10] Adam Grabowski. Auxiliary and approximating relations. Formalized Mathematics, 6(2):179-188, 1997.
• [11] L. Mirsky. A Dual of Dilworth's Decomposition Theorem. The American Mathematical Monthly, 78(8).
• [12] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
• [13] M. A. Perles. A Proof of Dilworth's Decomposition Theorem for Partially Ordered Sets. Israel Journal of Mathematics, 1:105-107, 1963.
• [14] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.
• [15] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.
• [16] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387-393, 1990.
• [17] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.