Sylvester–Gallai theorem
E571005
The Sylvester–Gallai theorem is a result in incidence geometry stating that for any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Sylvester–Gallai theorem canonical | 2 |
| Sylvester–Gallai configuration theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6149923 — 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: Sylvester–Gallai theorem Context triple: [James Joseph Sylvester, notableWork, Sylvester–Gallai 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.
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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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.
Erdős distinct distances problem
The Erdős distinct distances problem is a famous question in combinatorial geometry that asks for the minimum number of distinct distances determined by a given number of points in the plane.
- 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: Sylvester–Gallai theorem Target entity description: The Sylvester–Gallai theorem is a result in incidence geometry stating that for any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points.
-
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.
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.
-
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.
Erdős distinct distances problem
The Erdős distinct distances problem is a famous question in combinatorial geometry that asks for the minimum number of distinct distances determined by a given number of points in the plane.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
result in incidence geometry
ⓘ
theorem ⓘ |
| alsoKnownAs |
Sylvester–Gallai configuration theorem
ⓘ
linked to:
Sylvester–Gallai theorem
|
| appearsIn |
textbooks on combinatorial geometry
ⓘ
textbooks on discrete geometry ⓘ |
| appliesTo | finite sets of points in the Euclidean plane ⓘ |
| assumption |
not all points lie on a single line
ⓘ
the set of points is finite ⓘ |
| conclusion | there exists a line incident with exactly two of the points ⓘ |
| condition | the points are not all collinear ⓘ |
| doesNotRequire | points in general position ⓘ |
| field |
combinatorial geometry
ⓘ
discrete geometry ⓘ incidence geometry ⓘ |
| generalizationOf |
de Bruijn–Erdős theorem (incidence geometry)
ⓘ
linked to:
de Bruijn–Erdős theorem
|
| guaranteesExistenceOf | a line determined by exactly two points of the set ⓘ |
| hasConcept | ordinary line ⓘ |
| hasDimension | two-dimensional Euclidean space ⓘ |
| hasProofTechnique |
combinatorial arguments
ⓘ
geometric arguments ⓘ |
| holdsIn | Euclidean plane ⓘ |
| implies | existence of an ordinary line ⓘ |
| influenced |
development of incidence geometry
ⓘ
research on extremal configurations of points and lines ⓘ |
| isClassicalResult | true ⓘ |
| languageOfOriginalStatement | English ⓘ |
| mathematicsSubjectClassification | 52C10 ⓘ |
| namedAfter |
James Joseph Sylvester
ⓘ
Tibor Gallai ⓘ |
| proposedBy | James Joseph Sylvester ⓘ |
| provedBy | Tibor Gallai ⓘ |
| relatedConcept |
collinear points
ⓘ
finite geometry ⓘ incidence structure ⓘ |
| relatedTo |
Dirac–Motzkin conjecture
ⓘ
Erdős–de Bruijn theorem on incidences ⓘ
linked to:
de Bruijn–Erdős theorem
Kelly’s theorem on complex Sylvester–Gallai configurations ⓘ |
| statement | For any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points ⓘ |
| topic |
collinearity of points
ⓘ
finite point configurations ⓘ ordinary lines in point sets ⓘ |
| typeOfResult | existence theorem ⓘ |
| usedIn |
combinatorial incidence bounds
ⓘ
discrete geometry of the Euclidean plane ⓘ study of point-line arrangements ⓘ |
| yearProposed | 1893 ⓘ |
| yearProved | 1944 ⓘ |
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: Sylvester–Gallai theorem Description of subject: The Sylvester–Gallai theorem is a result in incidence geometry stating that for any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.