Tucker’s lemma

E83404

combinatorial lemma topological combinatorics result

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.

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All labels observed (5)

Label	Occurrences
Borsuk–Ulam theorem	4
Tucker’s lemma canonical	4
Borsuk–Ulam theorem (via combinatorial arguments)	1
Tucker's lemma	1
octahedral Tucker lemma	1

How this entity was disambiguated

This entity first appeared as the object of triple T677226 — 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: Tucker’s lemma
Context triple: [Albert W. Tucker, knownFor, Tucker’s lemma]

A. 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.
B. 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.
C. 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.
D. Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
E. Kalai–Smorodinsky bargaining solution
The Kalai–Smorodinsky bargaining solution is a cooperative game theory concept that selects a fair agreement between parties by preserving proportional gains relative to their best possible outcomes.
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: Tucker’s lemma
Target entity description: 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.

A. 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.
B. 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.
C. 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.
D. Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
E. Kalai–Smorodinsky bargaining solution
The Kalai–Smorodinsky bargaining solution is a cooperative game theory concept that selects a fair agreement between parties by preserving proportional gains relative to their best possible outcomes.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	combinatorial lemma ⓘ topological combinatorics result ⓘ
appliesTo	triangulated spheres ⓘ
assumes	antipodal symmetry of the triangulation ⓘ
concerns	Z2-equivariant combinatorial structures ⓘ antipodal labeling of triangulations ⓘ
field	combinatorics ⓘ topology ⓘ
generalizedBy	Ky Fan’s lemma ⓘ
guaranteesExistenceOf	complementary edge labels ⓘ
hasApplication	discrete versions of fair division theorems ⓘ
hasConclusion	existence of an edge whose endpoints have opposite labels ⓘ
hasConsequence	existence of complementary labeled simplex edges ⓘ
hasConstraint	labeling must be antipodal ⓘ labels exclude zero in the classical formulation ⓘ
hasDimensionParameter	n ⓘ
hasDomain	combinatorial topology ⓘ
hasInput	antipodally symmetric triangulation of a sphere ⓘ labeling of vertices by integers with opposite signs on antipodal points ⓘ
hasLabelSet	{±1,±2,…,±n} in the classical n-dimensional version ⓘ
hasNature	non-constructive existence result ⓘ
hasProofMethod	combinatorial methods ⓘ topological methods ⓘ
hasSpecialCase	discrete ham sandwich–type results ⓘ
hasVariant	Tucker–Fan type lemmas ⓘ cubical Tucker lemma ⓘ Tucker’s lemma self-linksurface differs ⓘ surface form: octahedral Tucker lemma
holdsOn	triangulations of the n-dimensional sphere ⓘ
implies	discrete Borsuk–Ulam type results ⓘ
isAnalogOf	Tucker’s lemma self-linksurface differs ⓘ surface form: Borsuk–Ulam theorem
isEquivalentTo	Borsuk–Ulam theorem over Z2 in appropriate formulations ⓘ
isToolFor	combinatorial fixed-point theory ⓘ discrete geometry ⓘ topological combinatorics ⓘ
namedAfter	Albert W. Tucker ⓘ
relatedTo	Ky Fan’s lemma ⓘ Sperner's lemma ⓘ surface form: Sperner’s lemma
usedFor	combinatorial proofs in consensus division ⓘ combinatorial proofs in fair division problems ⓘ combinatorial proofs in game theory ⓘ
usedIn	combinatorial proofs of fixed-point theorems ⓘ discrete versions of topological results ⓘ equivariant topology ⓘ proofs of the Borsuk–Ulam theorem ⓘ
usedToProve	consensus halving theorems ⓘ necklace splitting theorems ⓘ
yearIntroducedApprox	1940s ⓘ

How these facts were elicited

Referenced by (11)

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

Albert W. Tucker → knownFor → Tucker’s lemma ⓘ

Albert W. Tucker → notableWork → Tucker’s lemma ⓘ

Tucker’s lemma → isAnalogOf → Tucker’s lemma self-linksurface differs ⓘ

this entity surface form: Borsuk–Ulam theorem

Tucker’s lemma → hasVariant → Tucker’s lemma self-linksurface differs ⓘ

this entity surface form: octahedral Tucker lemma

Tucker → knownFor → Tucker’s lemma ⓘ

subject surface form: Albert W. Tucker

this entity surface form: Tucker's lemma

Sperner's lemma → usedForProofOf → Tucker’s lemma ⓘ

this entity surface form: Borsuk–Ulam theorem (via combinatorial arguments)

Ky Fan’s lemma → generalizes → Tucker’s lemma ⓘ

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

this entity surface form: Borsuk–Ulam theorem

Steinhaus chessboard theorem → relatedTo → Tucker’s lemma ⓘ

this entity surface form: Borsuk–Ulam theorem

Karol Borsuk → notableWork → Tucker’s lemma ⓘ

this entity surface form: Borsuk–Ulam theorem

Albert William Tucker → knownFor → Tucker’s lemma ⓘ

All labels observed (5)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (11)

How this entity was disambiguated

How these facts were elicited