Erdős distinct distances problem
E554301
The Erdős distinct distances problem is a famous question in combinatorial geometry that asks for the minimum number of distinct distances determined by a given number of points in the plane.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Erdős distinct distances problem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5896708 — 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 distinct distances problem Context triple: [Pál Erdős, knownFor, Erdős distinct distances problem]
-
A.
Erdős–Szekeres theorem
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.
-
B.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
C.
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.
-
D.
Helly’s theorem
Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
-
E.
Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
- 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 distinct distances problem Target entity description: The Erdős distinct distances problem is a famous question in combinatorial geometry that asks for the minimum number of distinct distances determined by a given number of points in the plane.
-
A.
Erdős–Szekeres theorem
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.
-
B.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
C.
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.
-
D.
Helly’s theorem
Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
-
E.
Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical problem
ⓘ
problem in combinatorial geometry ⓘ |
| alsoKnownAs | distinct distances problem NERFINISHED ⓘ |
| asksFor | minimum number of distinct distances determined by n points in the plane ⓘ |
| asymptoticForm | number of distinct distances is at least on the order of n / log n ⓘ |
| breakthroughMethod |
incidence geometry
ⓘ
polynomial method ⓘ |
| breakthroughYear | 2010 ⓘ |
| concerns | extremal configurations of points in the plane ⓘ |
| conjecturedOrderOriginally | n / sqrt(log n) ⓘ |
| difficulty | longstanding and hard problem in discrete geometry ⓘ |
| domain | Euclidean plane ⓘ |
| field |
combinatorial geometry
ⓘ
combinatorics ⓘ discrete geometry ⓘ |
| generalization |
distinct distances in higher dimensions
ⓘ
distinct distances in other metric spaces ⓘ |
| improvedLowerBound | c·n / log n ⓘ |
| improvedLowerBoundProvedBy | Larry Guth and Nets Hawk Katz NERFINISHED ⓘ |
| improvedLowerBoundYear | 2010 ⓘ |
| influenced |
applications of the polynomial method in combinatorics
ⓘ
development of incidence geometry ⓘ |
| involves | pairwise distances between points ⓘ |
| mainQuestion | Given n points in the plane, what is the minimum possible number of distinct pairwise distances? ⓘ |
| majorBreakthroughBy |
Larry Guth
NERFINISHED
ⓘ
Nets Hawk Katz NERFINISHED ⓘ |
| namedAfter | Paul Erdős NERFINISHED ⓘ |
| openAspect |
exact asymptotic constant in the lower bound
ⓘ
tight bound for all n ⓘ |
| originalLowerBound | c·n^{1/2} ⓘ |
| originalLowerBoundProposedBy | Paul Erdős NERFINISHED ⓘ |
| originalUpperBoundConstruction | n / sqrt(log n) ⓘ |
| originalUpperBoundConstructionBy | Paul Erdős NERFINISHED ⓘ |
| posedBy | Paul Erdős NERFINISHED ⓘ |
| relatedTo |
Szemerédi–Trotter theorem
NERFINISHED
ⓘ
incidence bounds between points and lines ⓘ polynomial partitioning technique ⓘ |
| statusBefore2010s | major open problem in combinatorial geometry ⓘ |
| typicalNotation | f(n) for the minimum number of distinct distances ⓘ |
| variable | n points in the plane ⓘ |
| yearPosed | 1946 ⓘ |
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 distinct distances problem Description of subject: The Erdős distinct distances problem is a famous question in combinatorial geometry that asks for the minimum number of distinct distances determined by a given number of points in the plane.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.