Erdős–Szekeres theorem

E386031

mathematical theorem theorem in combinatorial geometry

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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All labels observed (8)

Label	Occurrences
Erdős–Szekeres theorem canonical	7
Erdős–Szekeres conjecture	1
Erdős–Szekeres conjecture on ES(n)	1
Erdős–Szekeres convex polygon problem	1
Erdős–Szekeres monotone subsequence theorem	1
Erdős–Szekeres numbers	1
Erdős–Szekeres problem	1
Erdős–Szekeres problem on convex polygons	1

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]

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.
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
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.
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 conjecture on ES(n) ⓘ linked to: Erdős–Szekeres theorem
involvesConcept	binomial coefficients ⓘ convex position ⓘ extremal functions ⓘ general position of points ⓘ
lowerBound	ES(n) ≥ 2^{n-2} + 1 ⓘ
namedAfter	George Szekeres ⓘ Paul Erdős ⓘ linked to: Pál Erdős
openProblem	exact determination of ES(n) for general n ⓘ
originalAuthors	George Szekeres ⓘ Paul Erdős ⓘ linked to: Pál 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	Erdős–Szekeres number ⓘ linked to: Ramsey number
relatedTo	Erdős–Szekeres conjecture ⓘ linked to: Erdős–Szekeres theorem Erdős–Szekeres monotone subsequence theorem ⓘ linked to: Erdős–Szekeres 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-type theorem ⓘ linked to: Ramsey theory existence theorem ⓘ
usedIn	proofs in discrete geometry ⓘ results on convex position and order types ⓘ
yearProved	1935 ⓘ

How these facts were elicited

Referenced by (14)

Full triples — surface form annotated when it differs from this entity's canonical label.

George Szekeres → notableWork → Erdős–Szekeres theorem ⓘ

Pál Erdős → knownFor → Erdős–Szekeres theorem ⓘ

Pál Erdős → knownFor → Erdős–Szekeres convex polygon problem ⓘ