Erdős–Ko–Rado theorem
E554298
The Erdős–Ko–Rado theorem is a fundamental result in extremal combinatorics that determines the maximum size of a family of subsets of a finite set in which every pair of subsets has a non-empty intersection.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Erdős–Ko–Rado theorem canonical | 3 |
| Hilton–Milner theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5896705 — 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: Erdős–Ko–Rado theorem Context triple: [Pál Erdős, knownFor, Erdős–Ko–Rado theorem]
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A.
Sperner family
A Sperner family is a collection of subsets of a finite set in which no subset is contained within another, central in extremal set theory and combinatorics.
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B.
Erdős–Szekeres theorem
The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
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C.
de Bruijn–Erdős theorem
The de Bruijn–Erdős theorem is a fundamental result in combinatorics and graph theory that relates finite and infinite structures, notably asserting that certain properties of infinite graphs or set systems are determined by their finite substructures.
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D.
Radon’s theorem
Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
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E.
Helly’s theorem
Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
- 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: Erdős–Ko–Rado theorem Target entity description: The Erdős–Ko–Rado theorem is a fundamental result in extremal combinatorics that determines the maximum size of a family of subsets of a finite set in which every pair of subsets has a non-empty intersection.
-
A.
Sperner family
A Sperner family is a collection of subsets of a finite set in which no subset is contained within another, central in extremal set theory and combinatorics.
-
B.
Erdős–Szekeres theorem
The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
-
C.
de Bruijn–Erdős theorem
The de Bruijn–Erdős theorem is a fundamental result in combinatorics and graph theory that relates finite and infinite structures, notably asserting that certain properties of infinite graphs or set systems are determined by their finite substructures.
-
D.
Radon’s theorem
Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
-
E.
Helly’s theorem
Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
result in extremal combinatorics
ⓘ
theorem ⓘ |
| assumption | n ≥ 2k ⓘ |
| boundaryCase | for n = 2k, there exist non-star maximum intersecting families ⓘ |
| characterizes | maximum size of an intersecting family of k-subsets of [n] ⓘ |
| concerns |
families of k-element subsets
ⓘ
intersecting families of sets ⓘ maximum size of intersecting families ⓘ |
| defines | intersecting family as a family of sets in which every pair of sets has non-empty intersection ⓘ |
| domain | finite sets ⓘ |
| field |
combinatorics
ⓘ
extremal combinatorics ⓘ |
| hasApplication |
coding theory
ⓘ
design theory ⓘ graph theory ⓘ probabilistic combinatorics ⓘ |
| hasGeneralization |
Ahlswede–Khachatrian complete intersection theorem
NERFINISHED
ⓘ
Erdős–Ko–Rado-type theorems on hypergraphs NERFINISHED ⓘ Erdős–Ko–Rado-type theorems on permutations NERFINISHED ⓘ Erdős–Ko–Rado-type theorems on vector spaces NERFINISHED ⓘ Hilton–Milner theorem NERFINISHED ⓘ |
| hasProofMethod |
algebraic methods
ⓘ
compression method ⓘ graph-theoretic methods ⓘ shifting technique ⓘ |
| implies | any intersecting family of k-subsets of [n] with n ≥ 2k has size at most C(n−1, k−1) ⓘ |
| maximumAttainedBy | family of all k-subsets containing a fixed element ⓘ |
| namedAfter |
Chao Ko
NERFINISHED
ⓘ
Paul Erdős NERFINISHED ⓘ Richard Rado NERFINISHED ⓘ |
| originallyProvedBy |
Chao Ko
NERFINISHED
ⓘ
Paul Erdős NERFINISHED ⓘ Richard Rado NERFINISHED ⓘ |
| publishedIn | Journal of the London Mathematical Society NERFINISHED ⓘ |
| relatedTo |
Sperner's theorem
NERFINISHED
ⓘ
Turán-type extremal problems ⓘ intersection theorems ⓘ |
| statement | For n ≥ 2k, the largest size of an intersecting family of k-subsets of an n-element set is C(n−1, k−1). ⓘ |
| topic |
extremal set theory
NERFINISHED
ⓘ
intersection properties of set families ⓘ |
| typicalExtremalFamily | star family of k-subsets containing a fixed element ⓘ |
| uniquenessCondition | for n > 2k, the only maximum intersecting families are stars ⓘ |
| usesConcept |
binomial coefficients
ⓘ
intersecting set systems ⓘ k-uniform set systems ⓘ |
| yearProved | 1938 ⓘ |
| yearPublished | 1961 ⓘ |
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: Erdős–Ko–Rado theorem Description of subject: The Erdős–Ko–Rado theorem is a fundamental result in extremal combinatorics that determines the maximum size of a family of subsets of a finite set in which every pair of subsets has a non-empty intersection.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hilton–Milner theorem