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	2
polytopal Sperner lemma	2
Sperner boundary labeling condition	1
Sperner's lemma canonical	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's lemma self-linksurface differs ⓘ surface form: Sperner boundary labeling condition
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 ⓘ Sperner's lemma self-linksurface differs ⓘ surface form: polytopal Sperner 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 ⓘ Non-cooperative Games ⓘ surface form: Nash equilibrium Sperner family ⓘ
specialCaseOf	Sperner's lemma self-linksurface differs ⓘ surface form: polytopal Sperner lemma
typeOf	Sperner's lemma self-linksurface differs ⓘ surface form: combinatorial analog of Brouwer fixed-point theorem
typicalSetting	triangulation of an n-dimensional simplex ⓘ
usedForProofOf	Tucker’s lemma ⓘ surface form: Borsuk–Ulam theorem (via combinatorial arguments) 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 (7)

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 ⓘ

this entity surface form: Sperner’s lemma

Sperner's lemma → assumes → Sperner's lemma self-linksurface differs ⓘ

this entity surface form: Sperner boundary labeling condition

Sperner's lemma → specialCaseOf → Sperner's lemma self-linksurface differs ⓘ

this entity surface form: polytopal Sperner lemma

Sperner's lemma → typeOf → Sperner's lemma self-linksurface differs ⓘ

this entity surface form: combinatorial analog of Brouwer fixed-point theorem

Sperner's lemma → hasGeneralization → Sperner's lemma self-linksurface differs ⓘ