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Dilworth's theorem

In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of an antichain of incomparable elements equals the minimum number of chains needed to cover all elements. This number is called the width of the partial order. The theorem is named for the mathematician…

Art, Width of special partial orders & Perfection of comparability graphs

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Width of special partial orders

Boolean latticePower setInclusionSperner's theoremLubell–Yamamoto–Meshalkin inequality
9 related topics

Perfection of comparability graphs

Undirected graphCliqueIndependent setInduced subgraphPerfect
8 related topics

Extension to infinite partially ordered sets

ColoringIncomparability graphDe Bruijn–Erdős theoremCardinal numberא0
5 related topics

Proof via Kőnig's theorem

Kőnig's theoremBipartite graphHall's marriage theoremPolynomial time
4 related topics

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Statement

Inductive proof

Proof via Kőnig's theorem

Extension to infinite partially ordered sets

Dual of Dilworth's theorem (Mirsky's theorem)

Perfection of comparability graphs

Width of special partial orders

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Dilworth's theorem

Nodes46
Edges45
Triples157
Avg. degree1.96
Density0.043478
Components1

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Dilworth's theorem

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related to References · 98
Dilworth's theorem → Advances, Aldous, Algorithms, American Mathematical Monthly, American Mathematical Society, Annals, Applications, Berge, BF00143895, BF02759806, Chvátal, Claude, Combinatorial, Combinatorial Theory, Combinatorics, Curtis, Daniel, David, Decomposition Theorem, Diaconis
related to External links · 23
Dilworth's theorem → Babai, Borgersen, Combinatorics, Dilworth's Lemma, Equivalence, Eric, Graphs, Lecture, Lecture Notes, László, MathWorld, November, Orders, PDF, Perfect Graphs, PlanetMath, Probability, Raghavan, Recognition Algorithms, Robert
related to Perfection of comparability graphs · 11
Dilworth's theorem → An, Any, Berge, By, Chvátal, Dilworth's, Every, Lovász, Mirsky's, Therefore, Thus
related to Extension to infinite partially ordered sets · 8
Dilworth's theorem → By, De Bruijn, Dilworth's, Erdős, For, However, In, Therefore
related to Proof via Kőnig's theorem · 8
Dilworth's theorem → By Kőnig's, Dilworth's, Hall's, Kőnig's, Let, Like, Therefore, To
related to Dual of Dilworth's theorem (Mirsky's theorem) · 5
Dilworth's theorem → Dilworth's, Its, Mirsky's, Then, This
related to Statement · 4
Dilworth's theorem → An, Dilworth's, Here, The

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Important terminology

theorem antichain order displaystyle dilworth's chains size partial ordered set graph width partially chain number elements doi 10 decomposition perfect

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SubjectPredicateObjectConfidenceSrc
Dilworth's theoremrelated to Dual of Dilworth's theorem (Mirsky's theorem)Dilworth's0.60section
Dilworth's theoremrelated to Dual of Dilworth's theorem (Mirsky's theorem)This0.60section
Dilworth's theoremrelated to Dual of Dilworth's theorem (Mirsky's theorem)Mirsky's0.60section
Dilworth's theoremrelated to Dual of Dilworth's theorem (Mirsky's theorem)Its0.60section
Dilworth's theoremrelated to Dual of Dilworth's theorem (Mirsky's theorem)Then0.60section
Dilworth's theoremrelated to Extension to infinite partially ordered setsDilworth's0.60section
Dilworth's theoremrelated to Extension to infinite partially ordered setsFor0.60section
Dilworth's theoremrelated to Extension to infinite partially ordered setsBy0.60section
Dilworth's theoremrelated to Extension to infinite partially ordered setsTherefore0.60section
Dilworth's theoremrelated to Extension to infinite partially ordered setsDe Bruijn0.60section
Dilworth's theoremrelated to Extension to infinite partially ordered setsErdős0.60section
Dilworth's theoremrelated to Extension to infinite partially ordered setsHowever0.60section

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