Happy Ending problem
E386030
The Happy Ending problem is a famous combinatorial geometry question that investigates the minimum number of points in general position in the plane needed to guarantee the existence of a convex polygon with a given number of vertices.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Happy Ending problem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T3757267 — 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: Happy Ending problem Context triple: [George Szekeres, notableWork, Happy Ending problem]
-
A.
The Happy Ending
The Happy Ending is a 1969 American drama film written and directed by Richard Brooks, starring Jean Simmons as a disillusioned housewife who abruptly leaves her comfortable suburban life in search of independence and self-discovery.
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B.
"Happy Ending"
"Happy Ending" is a pop song by British singer-songwriter Mika, known for its emotive lyrics and soaring, melodic chorus.
-
C.
Conway's 99-graph problem
Conway's 99-graph problem is an unsolved combinatorial question in graph theory, posed by John H. Conway, concerning the existence and properties of a hypothetical 99-vertex graph with highly constrained adjacency conditions.
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D.
Happy Endings
Happy Endings is an American ensemble sitcom that follows a close-knit group of friends navigating relationships and adulthood in Chicago with fast-paced, joke-heavy humor.
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E.
Meeting Across the River
"Meeting Across the River" is a moody, jazz-tinged ballad by Bruce Springsteen that serves as a cinematic, character-driven prelude to "Jungleland" on the Born to Run album.
- 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: Happy Ending problem Target entity description: The Happy Ending problem is a famous combinatorial geometry question that investigates the minimum number of points in general position in the plane needed to guarantee the existence of a convex polygon with a given number of vertices.
-
A.
The Happy Ending
The Happy Ending is a 1969 American drama film written and directed by Richard Brooks, starring Jean Simmons as a disillusioned housewife who abruptly leaves her comfortable suburban life in search of independence and self-discovery.
-
B.
"Happy Ending"
"Happy Ending" is a pop song by British singer-songwriter Mika, known for its emotive lyrics and soaring, melodic chorus.
-
C.
Conway's 99-graph problem
Conway's 99-graph problem is an unsolved combinatorial question in graph theory, posed by John H. Conway, concerning the existence and properties of a hypothetical 99-vertex graph with highly constrained adjacency conditions.
-
D.
Happy Endings
Happy Endings is an American ensemble sitcom that follows a close-knit group of friends navigating relationships and adulthood in Chicago with fast-paced, joke-heavy humor.
-
E.
Meeting Across the River
"Meeting Across the River" is a moody, jazz-tinged ballad by Bruce Springsteen that serves as a cinematic, character-driven prelude to "Jungleland" on the Born to Run album.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical problem
ⓘ
problem in combinatorial geometry ⓘ problem in discrete geometry ⓘ |
| asksFor | minimum number of points in the plane in general position that guarantees a convex polygon with a given number of vertices ⓘ |
| assumes |
no three points are collinear
ⓘ
points are in general position ⓘ |
| basedOn | observation by Esther Klein ⓘ |
| concerns |
convex n-gons
ⓘ
existence of convex polygons ⓘ points in the Euclidean plane ⓘ |
| conjecture | f(n) = 2^{n-2} + 1 for all n ≥ 3 ⓘ |
| coreQuestion | for each integer n ≥ 3, determine the smallest number f(n) such that any set of f(n) points in general position in the plane contains n points in convex position ⓘ |
| difficulty | considered hard for large n ⓘ |
| field |
Ramsey theory
ⓘ
combinatorial geometry ⓘ discrete geometry ⓘ |
| firstPublishedIn | 1935 ⓘ |
| hasLowerBound | f(n) ≥ 2^{n-2} + 1 ⓘ |
| hasUpperBound | f(n) ≤ {2n-4 \choose n-2} + 1 (classical Erdős–Szekeres bound) ⓘ |
| hasVariant |
higher-dimensional versions asking for convex polytopes
ⓘ
problems with additional constraints on point sets ⓘ |
| improvedUpperBound | f(n) ≤ 2^{n+o(n)} ⓘ |
| influenced |
development of combinatorial geometry
ⓘ
research in geometric Ramsey theory ⓘ |
| introducedBy |
George Szekeres
ⓘ
Paul Erdős ⓘ
linked to:
Pál Erdős
|
| involvesConcept |
convex hull
ⓘ
extreme points ⓘ general position ⓘ |
| isAlsoKnownAs |
Erdős–Szekeres problem
ⓘ
linked to:
Erdős–Szekeres theorem
|
| knownResult |
f(3) = 3
ⓘ
f(4) = 5 ⓘ f(5) = 9 ⓘ f(6) = 17 ⓘ |
| namedAfter |
George Szekeres
ⓘ
Paul Erdős ⓘ
linked to:
Pál Erdős
|
| notation | f(n) ⓘ |
| originStory | name comes from the fact that George Szekeres and Esther Klein later married ⓘ |
| relatedPerson | Esther Klein ⓘ |
| relatedTo |
Erdős–Szekeres numbers
ⓘ
linked to:
Erdős–Szekeres theorem
Erdős–Szekeres theorem ⓘ Ramsey-type problems ⓘ convex position in the plane ⓘ |
| status |
open for general n
ⓘ
partially solved ⓘ |
| typicalExampleOf |
classical problem in discrete geometry
ⓘ
classical problem in extremal combinatorics ⓘ |
| yearProposed | 1933 ⓘ |
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: Happy Ending problem Description of subject: The Happy Ending problem is a famous combinatorial geometry question that investigates the minimum number of points in general position in the plane needed to guarantee the existence of a convex polygon with a given number of vertices.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.