Turán's theorem
E750082
Turán's theorem is a fundamental result in extremal graph theory that determines the maximum number of edges a graph can have without containing a complete subgraph of a given size.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Turán's theorem canonical | 2 |
| Turán’s theorem | 2 |
| Turán graph T_{r-1}(n) | 1 |
| Turán-type theorems for hypergraphs | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8669824 — 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: Turán's theorem Context triple: [Pál Turán, knownFor, Turán's theorem]
-
A.
Erdős–Stone theorem
The Erdős–Stone theorem is a fundamental result in extremal graph theory that asymptotically determines the maximum number of edges in an n-vertex graph that avoids containing a given subgraph.
-
B.
Erdős–Ko–Rado theorem
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.
-
C.
Erdős–Gallai theorem
The Erdős–Gallai theorem is a fundamental result in graph theory that characterizes which sequences of nonnegative integers can occur as the degree sequences of simple graphs.
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D.
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.
-
E.
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.
- 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: Turán's theorem Target entity description: Turán's theorem is a fundamental result in extremal graph theory that determines the maximum number of edges a graph can have without containing a complete subgraph of a given size.
-
A.
Erdős–Stone theorem
The Erdős–Stone theorem is a fundamental result in extremal graph theory that asymptotically determines the maximum number of edges in an n-vertex graph that avoids containing a given subgraph.
-
B.
Erdős–Ko–Rado theorem
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.
-
C.
Erdős–Gallai theorem
The Erdős–Gallai theorem is a fundamental result in graph theory that characterizes which sequences of nonnegative integers can occur as the degree sequences of simple graphs.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
result in extremal graph theory
ⓘ
theorem ⓘ |
| appearsIn |
combinatorics textbooks
ⓘ
introductory extremal graph theory courses ⓘ |
| assumes |
no loops
ⓘ
no multiple edges ⓘ |
| characterizes | extremal graphs without K_r ⓘ |
| dealsWith |
cliques
ⓘ
complete graphs ⓘ edge density ⓘ simple finite graphs ⓘ |
| edgeCountFormula | floor(((r-2)/(2(r-1))) * n^2) ⓘ |
| edgeCountType | exact formula ⓘ |
| extremalGraph |
Turán graph T_{r-1}(n)
ⓘ
linked to:
Turán's theorem
|
| extremalGraphProperty |
complete (r-1)-partite graph
ⓘ
parts as equal in size as possible ⓘ |
| field |
extremal graph theory
ⓘ
linked to:
extremal combinatorics
graph theory ⓘ |
| forbidsSubgraph | complete graph K_r ⓘ |
| generalizes | Mantel's theorem ⓘ |
| gives | exact extremal number ex(n, K_r) ⓘ |
| graphType | undirected graphs ⓘ |
| hasConsequence | stability results for near-extremal graphs ⓘ |
| hasExtension |
Turán-type theorems for hypergraphs
ⓘ
linked to:
Turán's theorem
stability versions of Turán's theorem ⓘ weighted versions of Turán's theorem ⓘ |
| implies | upper bounds on edge density avoiding K_r ⓘ |
| importance |
fundamental result in extremal graph theory
ⓘ
prototype of many extremal problems ⓘ |
| mainTopic | maximum number of edges in graphs without a given complete subgraph ⓘ |
| namedAfter | Pál Turán ⓘ |
| originalAuthor | Pál Turán ⓘ |
| originalPublicationLanguage |
Hungarian
ⓘ
linked to:
Hungarian language
|
| parameterizedBy |
clique size r
ⓘ
number of vertices n ⓘ |
| proofMethod |
averaging arguments
ⓘ
induction on the number of vertices ⓘ symmetrization ⓘ |
| relatedTo |
Erdős–Stone theorem
ⓘ
Mantel's theorem ⓘ Zarankiewicz problem ⓘ |
| specialCaseOf | extremal graph theory results ⓘ |
| statementInformal | Among all n-vertex graphs with no K_r subgraph, the Turán graph T_{r-1}(n) has the maximum number of edges ⓘ |
| usedIn |
Ramsey theory
ⓘ
bounding chromatic number via forbidden cliques ⓘ design of extremal constructions ⓘ probabilistic method in combinatorics ⓘ |
| yearProved | 1941 ⓘ |
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: Turán's theorem Description of subject: Turán's theorem is a fundamental result in extremal graph theory that determines the maximum number of edges a graph can have without containing a complete subgraph of a given size.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Turán's theorem
linked to: Turán's theorem
linked to: Turán's theorem
linked to: Turán's theorem