de Bruijn–Erdős theorem

E239169

mathematical theorem theorem in combinatorics theorem in graph theory

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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All labels observed (3)

Label	Occurrences
de Bruijn–Erdős theorem canonical	1
de Bruijn–Erdős theorem for hypergraphs	1
de Bruijn–Erdős theorem for set systems	1

How this entity was disambiguated

This entity first appeared as the object of triple T2169629 — 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: de Bruijn–Erdős theorem
Context triple: [N. G. de Bruijn, notableWork, de Bruijn–Erdős theorem]

A. 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.
B. Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
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. Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
E. 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.
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: de Bruijn–Erdős theorem
Target entity description: 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.

A. 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.
B. Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
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. Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
E. 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.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in combinatorics ⓘ theorem in graph theory ⓘ
appliesTo	hypergraphs ⓘ simple graphs ⓘ
classification	result about determination of infinite properties by finite substructures ⓘ
concerns	chromatic number ⓘ finite graphs ⓘ graph coloring ⓘ infinite graphs ⓘ set systems ⓘ
field	combinatorics ⓘ graph theory ⓘ
generalizes	finite graph coloring principles to infinite graphs ⓘ
hasConsequence	coloring properties of infinite graphs are determined by their finite subgraphs ⓘ many problems about infinite graphs reduce to problems about finite graphs ⓘ
hasProofMethod	combinatorial argument ⓘ compactness argument ⓘ topological methods ⓘ ultrafilter techniques ⓘ
hasVariant	de Bruijn–Erdős theorem self-linksurface differs ⓘ surface form: de Bruijn–Erdős theorem for hypergraphs de Bruijn–Erdős theorem self-linksurface differs ⓘ surface form: de Bruijn–Erdős theorem for set systems
implies	chromatic number of an infinite graph equals the supremum of chromatic numbers of its finite subgraphs ⓘ
involvesConcept	cardinality ⓘ finite subgraph ⓘ infinite graph ⓘ proper vertex coloring ⓘ
isFundamentalIn	structural graph theory ⓘ theory of infinite graph colorings ⓘ
namedAfter	N. G. de Bruijn ⓘ surface form: Nicolaas Govert de Bruijn Pál Erdős ⓘ surface form: Paul Erdős
originalPublication	N. G. de Bruijn and P. Erdős, A colour problem for infinite graphs and hypergraphs ⓘ
originalPublicationYear	1951 ⓘ
relatedTo	Ramsey's theorem ⓘ Tychonoff theorem for products of compact spaces ⓘ surface form: Tychonoff's theorem compactness theorem for first-order logic ⓘ
relates	finite structures ⓘ infinite structures ⓘ
statement	Every infinite graph with finite chromatic number has a finite subgraph with the same chromatic number ⓘ
typeOfResult	compactness-type theorem in combinatorics ⓘ
usedIn	Ramsey theory ⓘ combinatorial set theory ⓘ extremal combinatorics ⓘ infinite graph theory ⓘ
yearProved	1951 ⓘ

How these facts were elicited

Referenced by (3)

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

N. G. de Bruijn → notableWork → de Bruijn–Erdős theorem ⓘ

de Bruijn–Erdős theorem → hasVariant → de Bruijn–Erdős theorem self-linksurface differs ⓘ

this entity surface form: de Bruijn–Erdős theorem for hypergraphs

de Bruijn–Erdős theorem → hasVariant → de Bruijn–Erdős theorem self-linksurface differs ⓘ

this entity surface form: de Bruijn–Erdős theorem for set systems

All labels observed (3)

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Statements (45)

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Referenced by (3)

How this entity was disambiguated

How these facts were elicited