Sperner family
E518470
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Sperner family canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5425464 — 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 family Context triple: [Sperner's lemma, relatedTo, Sperner family]
-
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.
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.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
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D.
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.
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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: Sperner family Target entity description: 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.
-
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.
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.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
-
D.
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.
-
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 (48)
| Predicate | Object |
|---|---|
| instanceOf |
antichain of sets
ⓘ
combinatorial concept ⓘ set system ⓘ |
| achievesMaximumWhen |
family consists of all k-subsets with k = ceil(n/2)
ⓘ
family consists of all k-subsets with k = floor(n/2) ⓘ |
| alsoKnownAs |
Sperner collection
NERFINISHED
ⓘ
Sperner system NERFINISHED ⓘ antichain ⓘ |
| appearsIn | Sperner’s 1928 paper on subsets of a finite set NERFINISHED ⓘ |
| centralResult | Sperner theorem NERFINISHED ⓘ |
| constraintType | inclusion-free condition ⓘ |
| countingProblem | enumerate number of Sperner families on an n-element set ⓘ |
| definedOn | finite set ⓘ |
| field |
combinatorics
ⓘ
extremal set theory ⓘ |
| generalizationOf | antichain in any poset ⓘ |
| groundSetCardinality | finite ⓘ |
| hasVariant |
k-Sperner family
ⓘ
multiset Sperner family ⓘ vector Sperner family ⓘ |
| isClosedUnder | no nontrivial closure operations implied by definition ⓘ |
| keyProperty |
no set in the family is a subset of another set in the family
ⓘ
the family is an antichain under set inclusion ⓘ |
| mathematicalObjectType | family of subsets ⓘ |
| maximalityConcept | maximal Sperner family ⓘ |
| maxSizeOnNElementSet | binomial(n, floor(n/2)) ⓘ |
| namedAfter | Emanuel Sperner NERFINISHED ⓘ |
| optimizationQuestion | determine maximum size for given ground set size ⓘ |
| posetView | antichain in the poset (2^[n], subseteq) ⓘ |
| relatedConcept |
Boolean lattice
ⓘ
Dilworth theorem NERFINISHED ⓘ Erdos–Ko–Rado theorem NERFINISHED ⓘ LYM inequality NERFINISHED ⓘ antichain in a partially ordered set ⓘ chain decomposition ⓘ |
| relatedInequality | Lubell–Yamamoto–Meshalkin inequality NERFINISHED ⓘ |
| specialCaseOf | Sperner poset NERFINISHED ⓘ |
| structure | subset of the Boolean lattice ordered by inclusion ⓘ |
| studies | maximal size of antichains in the Boolean lattice ⓘ |
| subsetCondition | for all A,B in the family, A not subset of B and B not subset of A ⓘ |
| typicalNotation | F subset of 2^[n] ⓘ |
| typicalUniverse | power set of an n-element set ⓘ |
| underlyingRelation | set inclusion ⓘ |
| usedIn |
coding theory
ⓘ
design theory ⓘ extremal combinatorics NERFINISHED ⓘ information theory ⓘ probabilistic 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: Sperner family Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.