Sperner's lemma

E121351

combinatorial theorem mathematical lemma result in combinatorial topology

Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.

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All labels observed (5)

Label	Occurrences
Sperner's lemma canonical	5
Sperner’s lemma	3
polytopal Sperner lemma	2
Sperner boundary labeling condition	1
combinatorial analog of Brouwer fixed-point theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T1056923 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Sperner's lemma
Context triple: [Brouwer fixed-point theorem, hasCombinatorialVersion, Sperner's lemma]

A. Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
B. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
C. Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
D. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
E. Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Sperner's lemma
Target entity description: Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.

A. Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
B. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
C. Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
D. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
E. Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	combinatorial theorem ⓘ mathematical lemma ⓘ result in combinatorial topology ⓘ
appearsIn	combinatorics textbooks ⓘ game theory textbooks ⓘ undergraduate topology textbooks ⓘ
appliesTo	labeled triangulations of simplices ⓘ triangulated simplices with boundary conditions ⓘ
assumes	Sperner boundary labeling condition ⓘ linked to: Sperner's lemma
conclusion	the number of fully labeled simplices is odd ⓘ there exists at least one fully labeled simplex ⓘ
coreConcept	boundary conditions determine interior structure ⓘ labeling of vertices of a triangulation ⓘ
dimension	holds in any finite dimension ⓘ
enables	discrete approximation of fixed points ⓘ
field	combinatorial topology ⓘ combinatorics ⓘ topology ⓘ
guaranteesExistenceOf	fully labeled simplex ⓘ
hasCombinatorialNature	yes ⓘ
hasGeneralization	Tucker's lemma ⓘ polytopal Sperner lemma ⓘ linked to: Sperner's lemma
hasProofMethod	induction on dimension ⓘ parity argument ⓘ
implies	existence of a panchromatic simplex ⓘ
influenced	development of combinatorial fixed-point theory ⓘ
isConstructive	yes ⓘ
labelingRule	vertices on a face may only use labels of that face ⓘ
namedAfter	Emanuel Sperner ⓘ
relatedTo	Brouwer fixed-point theorem ⓘ Knaster–Kuratowski–Mazurkiewicz lemma ⓘ Nash equilibrium ⓘ linked to: Non-cooperative Games Sperner family ⓘ
specialCaseOf	polytopal Sperner lemma ⓘ linked to: Sperner's lemma
typeOf	combinatorial analog of Brouwer fixed-point theorem ⓘ linked to: Sperner's lemma
typicalSetting	triangulation of an n-dimensional simplex ⓘ
usedForProofOf	Borsuk–Ulam theorem (via combinatorial arguments) ⓘ linked to: Tucker’s lemma Brouwer fixed-point theorem ⓘ Kakutani fixed-point theorem ⓘ existence of Nash equilibria ⓘ existence of economic equilibria ⓘ
usedIn	algorithmic game theory ⓘ combinatorial proofs of fixed-point theorems ⓘ computational topology ⓘ fair division problems ⓘ
usedInComplexityTheory	PPAD-completeness results ⓘ
yearProved	1928 ⓘ

How these facts were elicited

Referenced by (12)

Full triples — surface form annotated when it differs from this entity's canonical label.

Brouwer fixed-point theorem → hasCombinatorialVersion → Sperner's lemma ⓘ

Tucker’s lemma → relatedTo → Sperner’s lemma ⓘ

linked to: Sperner's lemma

Sperner's lemma → assumes → Sperner boundary labeling condition ⓘ

linked to: Sperner's lemma

Sperner's lemma → specialCaseOf → polytopal Sperner lemma ⓘ

linked to: Sperner's lemma

Sperner's lemma → typeOf → combinatorial analog of Brouwer fixed-point theorem ⓘ

linked to: Sperner's lemma

Sperner's lemma → hasGeneralization → polytopal Sperner lemma ⓘ

linked to: Sperner's lemma

Ky Fan’s lemma → relatedTo → Sperner’s lemma ⓘ

linked to: Sperner's lemma

Emanuel Sperner → notableWork → Sperner's lemma ⓘ

Emanuel Sperner → notableConcept → Sperner's lemma ⓘ

Knaster–Kuratowski–Mazurkiewicz lemma → relatedTo → Sperner's lemma ⓘ

Scarf algorithm → basedOn → Sperner's lemma ⓘ

Scarf’s lemma → relatedTo → Sperner’s lemma ⓘ

linked to: Sperner's lemma

All labels observed (5)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (12)

How this entity was disambiguated

How these facts were elicited