Conway's thrackle conjecture
E266111
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Conway's thrackle conjecture canonical | 1 |
| thrackle conjecture | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2426179 — 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: Conway's thrackle conjecture Context triple: [John H. Conway, hasConcept, Conway's thrackle conjecture]
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A.
Conway’s Game of Sprouts
Conway’s Game of Sprouts is a pencil-and-paper topological game in which players alternately connect dots with lines under simple rules, leading to rich combinatorial and mathematical analysis.
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B.
Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
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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.
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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E.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
- 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: Conway's thrackle conjecture Target entity description: 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.
-
A.
Conway’s Game of Sprouts
Conway’s Game of Sprouts is a pencil-and-paper topological game in which players alternately connect dots with lines under simple rules, leading to rich combinatorial and mathematical analysis.
-
B.
Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
-
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.
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.
-
E.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
problem in combinatorial geometry ⓘ unsolved problem in mathematics ⓘ |
| alsoKnownAs |
Conway's thrackle conjecture
ⓘ
surface form:
thrackle conjecture
|
| ambientSpace |
Euclidean space
ⓘ
surface form:
Euclidean plane
topological sphere ⓘ |
| appearsIn |
research literature on thrackles
ⓘ
surveys on open problems in combinatorial geometry ⓘ |
| concerns |
drawings of graphs in the plane
ⓘ
relationship between number of edges and vertices ⓘ thrackles ⓘ |
| conjectureYear | 1969 ⓘ |
| difficulty | longstanding open problem ⓘ |
| domain | finite simple graphs ⓘ |
| edgeIntersectionCondition | every pair of edges meets exactly once ⓘ |
| edgeIntersectionType | intersection may be at a common endpoint or at a proper crossing ⓘ |
| field |
combinatorial geometry
ⓘ
graph theory ⓘ |
| formulatedBy |
John H. Conway
ⓘ
surface form:
John Horton Conway
|
| hasKeyTerm |
crossing
ⓘ
edge ⓘ graph drawing ⓘ intersection ⓘ vertex ⓘ |
| hasSpecialCase |
conjecture for bipartite graphs
ⓘ
conjecture for cycles ⓘ conjecture for planar thrackles ⓘ |
| implies |
complete graphs with many edges cannot be drawn as thrackles
ⓘ
no thrackle drawing of a graph can have more edges than vertices ⓘ |
| inequalityForm | m ≤ n for thrackles, where m is number of edges and n is number of vertices ⓘ |
| knownResult |
linear upper bounds on the number of edges in a thrackle have been established
ⓘ
the conjecture is proved for several special classes of graphs ⓘ upper bounds better than 2n for the number of edges in a thrackle are known ⓘ |
| motivation | understanding extremal properties of graph drawings ⓘ |
| namedAfter |
John H. Conway
ⓘ
surface form:
John Horton Conway
|
| objectType | conjecture about edge–vertex bounds in graph drawings ⓘ |
| openAspect | whether the optimal upper bound on the number of edges in a thrackle is exactly the number of vertices ⓘ |
| relatedConcept |
crossing number
ⓘ
geometric graph theory ⓘ pseudoline arrangement ⓘ thrackle ⓘ topological graph ⓘ |
| researchArea |
extremal graph theory
ⓘ
topological graph theory ⓘ |
| statement | in any thrackle drawing of a finite graph, the number of edges is at most the number of vertices ⓘ |
| status | open ⓘ |
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: Conway's thrackle conjecture Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
John H. Conway
this entity surface form:
thrackle conjecture