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Dilworth's theorem: Art, Width of special partial orders & Perfection of comparability graphs

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…

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Dilworth's theorem topic overview

The analysis highlights Art, Width of special partial orders and Perfection of comparability graphs as prominent areas in the source structure around Dilworth's theorem.

Related topics
37
Source areas
8
Connected nodes
45
Extracted relationships
157
Concept neighborhoods
31
Bridge connections
45

What this topic covers Research coverage

Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.

Width of special partial orders · 9 topics
Perfection of comparability graphs · 8 topics
Overview · 7 topics
Extension to infinite partially ordered sets · 5 topics
Proof via Kőnig's theorem · 4 topics
Inductive proof · 2 topics
Dual of Dilworth's theorem (Mirsky's theorem) · 1 topics
Statement · 1 topics

Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.

Explore all related topics Closing gaps

Browse the complete topic structure, not only the most central items. Less prominent entities and concepts can reveal missing angles, specialized context and useful research gaps. Each item opens a new analysis centered on that subject.

Overview

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

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

How Dilworth's theorem connects Entity context

The extracted context around Dilworth's theorem shows recurring relationship patterns in the source. For example, 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 Another extracted example is 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. Use these groups to spot repeated connection types before inspecting the individual relationships.

Dilworth's theorem

Top relations

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

Important terminology

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

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

Dilworth's theorem relationships Subject–Predicate–Object triples

TTTA extracted 157 structured relationships around Dilworth's theorem. Examples in this analysis include Dilworth's theorem → related to Dual of Dilworth's theorem (Mirsky's theorem) → Dilworth's and Dilworth's theorem → related to Dual of Dilworth's theorem (Mirsky's theorem) → This. The table shows each extracted connection, where it came from and its confidence.

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

Related concept clusters Concept neighborhoods

The concept neighborhoods around Dilworth's theorem bring nearby vocabulary together. In this analysis, examples include Theorem, Partial and Width. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Dilworth's theorem
    • Theorem
    • Partial
    • Width
    • Finite
    • Order
    • Kőnig's
    • Partially
    • Number
    • Graphs
    • Proof
    • States
    • Ordered
  • dilworth's theorem
    • Theorem
    • Graph
    • Partial
    • Width
    • Finite
    • Order
    • Kőnig's
    • Partially
    • Number
    • Graphs
    • Proof
    • States
  • mathematics
    • Doi
    • Ordered
    • Sets
    • Perfect
    • Partially
    • Dilworth's
    • Number
    • Theorem
    • Jstor
    • Combinatorics
    • Mr
    • Graphs
  • order theory
    • Partial
    • Number
    • Width
    • Chain
    • Graph
    • Size
    • Theorem
    • Edge
    • Partition
    • Largest
    • Kőnig's
    • Orders
  • partially ordered set
    • Ordered
    • Partially
    • Set
    • Sets
    • Infinite
    • States
    • Decomposition
    • Size
    • Theorem
    • Width
    • Chain
    • Jstor
  • antichain
    • Size
    • Chain
    • Chains
    • Set
    • Displaystyle
    • Largest
    • Elements
    • Partially
    • Number
    • States
    • Order
    • Ordered
  • chains
    • Partition
    • Size
    • Chain
    • Disjoint
    • Elements
    • Partially
    • Number
    • Ordered
    • Order
    • Decomposition
    • Theorem
    • Exists
  • order dimension
    • Partial
    • Number
    • Width
    • Chain
    • Graph
    • Size
    • Theorem
    • Edge
    • Partition
    • Largest
    • Kőnig's
    • Orders

Connections between topic areas Semantic bridges

For Dilworth's theorem, one of the stronger structural bridges in this analysis connects Dilworth's theorem with Width of special partial orders. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.

Min side: 3
Dilworth's theoremWidth of special partial orders · splits 36 ⟂ 10
Dilworth's theoremPerfection of comparability graphs · splits 37 ⟂ 9
Dilworth's theoremOverview · splits 38 ⟂ 8
Dilworth's theoremExtension to infinite partially ordered sets · splits 40 ⟂ 6
Dilworth's theoremProof via Kőnig's theorem · splits 41 ⟂ 5
Dilworth's theoremInductive proof · splits 43 ⟂ 3

Map overview Semantic statistics

Dilworth's theorem

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

Source & methodology

TTTA analyzes the structure around Dilworth's theorem to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as Art, Width of special partial orders & Perfection of comparability graphs, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Dilworth's theorem · EN edition · Analysis: TopicsToTalkAbout

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