Ky Fan’s lemma
E321096
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Ky Fan’s lemma canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T3044200 — 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: Ky Fan’s lemma Context triple: [Tucker’s lemma, relatedTo, Ky Fan’s lemma]
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A.
Sperner's lemma
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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B.
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.
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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.
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D.
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.
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E.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
- 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: Ky Fan’s lemma Target entity description: Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
-
A.
Sperner's lemma
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.
-
B.
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.
-
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.
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.
-
E.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
- F. None of above. chosen
Statements (29)
| Predicate | Object |
|---|---|
| instanceOf |
combinatorial topological result
ⓘ
lemma in topology ⓘ mathematical theorem ⓘ |
| appliesTo |
labeled triangulations of simplices
ⓘ
labeled triangulations of spheres ⓘ |
| assumes |
labeling satisfying certain sign or parity constraints
ⓘ
triangulation with antipodal symmetry conditions ⓘ |
| concerns |
antipodal labelings
ⓘ
parity arguments in triangulations ⓘ |
| ensures | presence of a simplex with labels forming a prescribed pattern ⓘ |
| field |
combinatorial topology
ⓘ
combinatorics ⓘ topology ⓘ |
| generalizes | Tucker’s lemma ⓘ |
| guaranteesExistenceOf |
balanced simplices
ⓘ
fully labeled simplices ⓘ |
| hasVersion |
simplicial version
ⓘ
spherical version ⓘ |
| implies | Tucker’s lemma in special cases ⓘ |
| namedAfter | Ky Fan ⓘ |
| relatedTo |
Tucker’s lemma
ⓘ
surface form:
Borsuk–Ulam theorem
Sperner's lemma ⓘ
surface form:
Sperner’s lemma
|
| topicOf |
expositions in topological combinatorics textbooks
ⓘ
research in combinatorial fixed-point theory ⓘ |
| typeOfConclusion | existence theorem ⓘ |
| usedIn |
combinatorial proofs of fixed-point theorems
ⓘ
equivariant topology ⓘ fair division problems ⓘ topological combinatorics ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Ky Fan’s lemma Description of subject: Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.