Erdős–Szekeres theorem
E386031
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.
All labels observed (8)
How this entity was disambiguated
This entity first appeared as the object of triple T3757268 — 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–Szekeres theorem Context triple: [George Szekeres, notableWork, Erdős–Szekeres 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.
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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C.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
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D.
Conway's thrackle conjecture
Conway's thrackle conjecture is an unsolved problem in combinatorial geometry asserting that in any drawing of a graph where every pair of edges meets exactly once, the number of edges cannot exceed the number of vertices.
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E.
Conway circle theorem
The Conway circle theorem is a geometric result in triangle geometry that identifies a special circle associated with a triangle and certain constructed points, revealing notable collinearities and concyclicity relationships.
- 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–Szekeres theorem Target entity description: 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.
-
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.
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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C.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
-
D.
Conway's thrackle conjecture
Conway's thrackle conjecture is an unsolved problem in combinatorial geometry asserting that in any drawing of a graph where every pair of edges meets exactly once, the number of edges cannot exceed the number of vertices.
-
E.
Conway circle theorem
The Conway circle theorem is a geometric result in triangle geometry that identifies a special circle associated with a triangle and certain constructed points, revealing notable collinearities and concyclicity relationships.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in combinatorial geometry ⓘ |
| alsoKnownAs | happy ending theorem ⓘ |
| appliesTo | finite point sets in the Euclidean plane ⓘ |
| assumption |
no three points are collinear
ⓘ
points are in general position ⓘ |
| citedAs | classical result in combinatorial geometry ⓘ |
| conclusion | existence of n points in convex position ⓘ |
| exactValueKnownFor | small values of n ⓘ |
| field |
combinatorial geometry
ⓘ
combinatorics ⓘ discrete geometry ⓘ |
| guaranteesExistenceOf | convex k-gon ⓘ |
| hasApplication |
computational geometry
ⓘ
geometric Ramsey theory ⓘ theory of order types of point sets ⓘ |
| hasGeneralization | higher-dimensional variants for convex polytopes ⓘ |
| historicalNote | nicknamed happy ending theorem because the problem led to the marriage of George Szekeres and Esther Klein ⓘ |
| inspired |
Erdős–Szekeres theorem
self-linksurface differs
ⓘ
surface form:
Erdős–Szekeres conjecture on ES(n)
|
| involvesConcept |
binomial coefficients
ⓘ
convex position ⓘ extremal functions ⓘ general position of points ⓘ |
| lowerBound | ES(n) ≥ 2^{n-2} + 1 ⓘ |
| namedAfter |
George Szekeres
ⓘ
Pál Erdős ⓘ
surface form:
Paul Erdős
|
| openProblem | exact determination of ES(n) for general n ⓘ |
| originalAuthors |
George Szekeres
ⓘ
Pál Erdős ⓘ
surface form:
Paul Erdős
|
| originalBound | ES(n) ≤ \binom{2n-4}{n-2} + 1 ⓘ |
| originalResult | for every integer n ≥ 3 there exists a minimum number ES(n) such that any set of at least ES(n) points in general position in the plane contains n points in convex position ⓘ |
| publishedIn | Compositio Mathematica ⓘ |
| relatedConcept |
Ramsey number
ⓘ
surface form:
Erdős–Szekeres number
|
| relatedTo |
Erdős–Szekeres theorem
self-linksurface differs
ⓘ
surface form:
Erdős–Szekeres conjecture
Erdős–Szekeres theorem self-linksurface differs ⓘ
surface form:
Erdős–Szekeres monotone subsequence theorem
Ramsey theory ⓘ |
| requires | sufficiently large number of points ⓘ |
| statementInformal | any sufficiently large set of points in the plane in general position contains the vertices of a large convex polygon ⓘ |
| topic |
Ramsey-type results in geometry
ⓘ
convex polygons ⓘ extremal combinatorics ⓘ |
| type |
Ramsey theory
ⓘ
surface form:
Ramsey-type theorem
existence theorem ⓘ |
| usedIn |
proofs in discrete geometry
ⓘ
results on convex position and order types ⓘ |
| yearProved | 1935 ⓘ |
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–Szekeres theorem Description of subject: 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.
Referenced by (13)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Erdős–Szekeres convex polygon problem
subject surface form:
George Szekeres
this entity surface form:
Erdős–Szekeres problem
this entity surface form:
Erdős–Szekeres numbers
this entity surface form:
Erdős–Szekeres conjecture on ES(n)
this entity surface form:
Erdős–Szekeres monotone subsequence theorem
this entity surface form:
Erdős–Szekeres conjecture
this entity surface form:
Erdős–Szekeres problem on convex polygons