Erdős–Stone theorem
E554300
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Erdős–Stone theorem canonical | 3 |
| Erdős–Stone–Simonovits theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5896707 — 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–Stone theorem Context triple: [Pál Erdős, knownFor, Erdős–Stone theorem]
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A.
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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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.
Ramsey theory
Ramsey theory is a branch of combinatorics that studies the conditions under which order or structure must appear within sufficiently large or complex mathematical objects.
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E.
Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
- 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–Stone theorem Target entity description: 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.
-
A.
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.
-
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.
Ramsey theory
Ramsey theory is a branch of combinatorics that studies the conditions under which order or structure must appear within sufficiently large or complex mathematical objects.
-
E.
Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
result in extremal graph theory
ⓘ
theorem ⓘ |
| appliesTo |
finite simple graphs
ⓘ
n-vertex graphs ⓘ |
| assumes | fixed forbidden graph H independent of n ⓘ |
| characterizes | maximum number of edges in an n-vertex graph avoiding a fixed subgraph ⓘ |
| classification | cornerstone of modern extremal combinatorics ⓘ |
| concerns |
Turán-type extremal problems
ⓘ
asymptotic edge density ⓘ extremal number of edges in graphs ⓘ forbidden subgraphs ⓘ |
| domain | n → ∞ asymptotic regime ⓘ |
| field |
extremal graph theory
ⓘ
graph theory ⓘ |
| generalizes |
Turán’s theorem
ⓘ
linked to:
Turán's theorem
|
| givesAsymptoticsFor | extremal function ex(n,H) ⓘ |
| hasVariant |
Erdős–Stone–Simonovits theorem
ⓘ
linked to:
Erdős–Stone theorem
|
| implies |
extremal edge density depends only on chromatic number of forbidden graph for non-bipartite H
ⓘ
graphs with many edges contain large complete multipartite subgraphs ⓘ |
| influenced | development of extremal graph theory ⓘ |
| involvesConcept |
chromatic number χ(H)
ⓘ
edge density ⓘ forbidden subgraph H ⓘ o(1) term in asymptotics ⓘ |
| isDescribedAs |
asymptotic solution to general Turán-type problems for non-bipartite graphs
ⓘ
fundamental result in extremal graph theory ⓘ |
| namedAfter |
Arthur H. Stone
ⓘ
Paul Erdős ⓘ
linked to:
Pál Erdős
|
| originalAuthors |
Arthur H. Stone
ⓘ
Paul Erdős ⓘ
linked to:
Pál Erdős
|
| publishedIn | Proceedings of the London Mathematical Society ⓘ |
| relatedConcept | complete (χ(H)-1)-partite Turán graph ⓘ |
| relatesTo |
Turán’s theorem
ⓘ
linked to:
Turán's theorem
chromatic number of a graph ⓘ |
| saysAsymptoticallyEquivalentTo | edge number of Turán graph T_{χ(H)-1}(n) for non-bipartite H ⓘ |
| statesThat | for every non-bipartite graph H, ex(n,H) = (1 - 1/(χ(H)-1) + o(1)) * n^2 / 2 ⓘ |
| topic | forbidden subgraph problems in dense graphs ⓘ |
| typeOfResult | asymptotic extremal bound ⓘ |
| usedFor |
estimating extremal numbers for non-bipartite forbidden graphs
ⓘ
proving existence of dense substructures in graphs ⓘ |
| yearProved | 1946 ⓘ |
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–Stone theorem Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Erdős–Stone theorem