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.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
result in incidence geometry
ⓘ
theorem ⓘ |
| alsoKnownAs | Sylvester–Gallai configuration theorem NERFINISHED ⓘ |
| 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) NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Tibor Gallai NERFINISHED ⓘ |
| proposedBy | James Joseph Sylvester NERFINISHED ⓘ |
| provedBy | Tibor Gallai NERFINISHED ⓘ |
| relatedConcept |
collinear points
ⓘ
finite geometry ⓘ incidence structure ⓘ |
| relatedTo |
Dirac–Motzkin conjecture
NERFINISHED
ⓘ
Erdős–de Bruijn theorem on incidences NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.