In graph theory, the De Bruijn–Erdős theorem relates graph coloring of an infinite graph to the same problem on its finite subgraphs. It states that, when all finite subgraphs can be colored with
c
The De Bruijn–Erdős theorem has several different proofs, all depending in some way on the axiom of choice. Its applications include extending the four-color theorem and Dilworth's theorem from finite graphs and partially ordered sets to infinite ones, and reducing the Hadwiger–Nelson problem on the chromatic number of the plane to a problem about finite graphs. It may be generalized from finite numbers of colors to sets of colors whose cardinality is a strongly compact cardinal.
An undirected graph is a mathematical object consisting of a set of vertices and a set of edges that link pairs of vertices. The two vertices associated with each edge are called its endpoints. The graph is finite when its vertices and edges form finite sets, and infinite otherwise. A graph coloring associates each vertex with a color drawn from a set of colors, in such a way that every edge has two different colors at its endpoints. A frequent goal in graph coloring is to minimize the total number of colors that are used; the chromatic number of a graph is this minimum number of colors.[1] The four-color theorem states that every finite graph that can be drawn without crossings in the Euclidean plane needs at most four colors; however, some graphs with more complicated connectivity require more than four colors.[2] It is a consequence of the axiom of choice that the chromatic number is well-defined for infinite graphs, but for these graphs the chromatic number might itself be an infinite cardinal number.
A subgraph of a graph is another graph obtained from a subset of its vertices and a subset of its edges. If the larger graph is colored, the same coloring can be used for the subgraph. Therefore, the chromatic number of a subgraph cannot be larger than the chromatic number of the whole graph. The De Bruijn–Erdős theorem concerns the chromatic numbers of infinite graphs, and shows that (again, assuming the axiom of choice) they can be calculated from the chromatic numbers of their finite subgraphs. It states that, if the chromatic numbers of the finite subgraphs of a graph
G
c
G
c
G
G
The original motivation of Erdős in studying this problem was to extend from finite to infinite graphs the theorem that, whenever a graph has an orientation with finite maximum out-degree
k
(2k+1)
2k
2k+1
k=1
(2k+1)
Another application of the De Bruijn–Erdős theorem is to the Hadwiger–Nelson problem, which asks how many colors are needed to color the points of the Euclidean plane so that every two points that are a unit distance apart have different colors. This is a graph coloring problem for an infinite graph that has a vertex for every point of the plane and an edge for every two points whose Euclidean distance is exactly one. The induced subgraphs of this graph are called unit distance graphs. A seven-vertex unit distance graph, the Moser spindle, requires four colors; in 2018, much larger unit distance graphs were found that require five colors. The whole infinite graph has a known coloring with seven colors based on a hexagonal tiling of the plane. Therefore, the chromatic number of the plane must be either 5, 6, or 7, but it is not known which of these three numbers is the correct value. The De Bruijn–Erdős theorem shows that, for this problem, there exists a finite unit distance graph with the same chromatic number as the whole plane, so if the chromatic number is greater than five then this fact can be proved by a finite calculation.
The De Bruijn–Erdős theorem may also be used to extend Dilworth's theorem from finite to infinite partially ordered sets. Dilworth's theorem states that the width of a partial order (the maximum number of elements in a set of mutually incomparable elements) equals the minimum number of chains (totally ordered subsets) into which the partial order may be partitioned. A partition into chains may be interpreted as a coloring of the incomparability graph of the partial order. This is a graph with a vertex for each element of the order and an edge for each pair of incomparable elements. Using this coloring interpretation, together with a separate proof of Dilworth's theorem for finite partially ordered sets, it is possible to prove that an infinite partially ordered set has finite width
w
w
In the same way, the De Bruijn–Erdős theorem extends the four-color theorem from finite planar graphs to infinite planar graphs. Every finite planar graph can be colored with four colors, by the four-color theorem. The De Bruijn–Erdős theorem then shows thatevery graph that can be drawn without crossings in the plane, finite or infinite, can be colored with four colors. More generally, every infinite graph for which all finite subgraphs are planar can again be four-colored.[6]
The original proof of the De Bruijn–Erdős theorem, by De Bruijn, used transfinite induction.
provided the following very short proof, based on Tychonoff's compactness theorem in topology. Suppose that, for the given infinite graph
G
k
X
k
G
X
kV(G)
V(G)
F
G
XF
X
F
XF
G
A different proof using Zorn's lemma was given by Lajos Pósa, and also in the 1951 Ph.D. thesis of Gabriel Andrew Dirac. If
G
k
M
k
M
k
G
The theorem can also be proved using ultrafilters or non-standard analysis. gives a proof for graphs with a countable number of vertices based on Kőnig's infinity lemma.
All proofs of the De Bruijn–Erdős theorem use some form of the axiom of choice. Some form of this assumption is necessary, as there exist models of mathematics in which both the axiom of choice and the De Bruijn–Erdős theorem are false. More precisely, showed that the theorem is a consequence of the Boolean prime ideal theorem, a property that is implied by the axiom of choice but weaker than the full axiom of choice, and showed that the De Bruijn–Erdős theorem and the Boolean prime ideal theorem are equivalent in axiomatic power.[9] The De Bruijn–Erdős theorem for countable graphs can also be shown to be equivalent in axiomatic power, within a certain theory of second-order arithmetic, to Weak Kőnig's lemma.
For a counterexample to the theorem in models of set theory without choice, let
G
G
x
y
|x-y|\pm\sqrt2
x
x+q\pm\sqrt2
q
x
x
\sqrt2
x
\sqrt2
G
G
G
G
proves the following theorem, which may be seen as a generalization of the De Bruijn–Erdős theorem. Let
V
v
V
cv
S
V
CS
S
v
S
cv
\chi
V
S
T
\chi
CT
k
G
k
G
If a graph does not have finite chromatic number, then the De Bruijn–Erdős theorem implies that it must contain finite subgraphs of every possible finite chromatic number. Researchers have also investigated other conditions on the subgraphs that are forced to occur in this case. For instance, unboundedly chromatic graphs must also contain every possible finite bipartite graph as a subgraph. However, they may have arbitrarily large odd girth, and therefore they may avoid any finite set of non-bipartite subgraphs.[12]
The De Bruijn–Erdős theorem also applies directly to hypergraph coloring problems, where one requires that each hyperedge have vertices of more than one color. As for graphs, a hypergraph has a
k
k
The theorem may also be generalized to situations in which the number of colors is an infinite cardinal number. If
\kappa
G
\mu<\kappa
G
\mu
\kappa
\mu
\kappa=\aleph0
\aleph0
\gamma
G
(2\gamma)+
G
\gamma
G
G
\gamma
\kappa
\kappa
\kappa