# Teorema di De Bruijn-Erdős (teoria dei grafi) Nella teoria dei grafi, the De Bruijn–Erdős theorem relates graph coloring of an infinite graph to the same problem on its finite subgraphs. Lo afferma, when all finite subgraphs can be colored with {stile di visualizzazione c} colors, the same is true for the whole graph. The theorem was proved by Nicolaas Govert de Bruijn and Paul Erdős (1951), da cui prende il nome.

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.

Contenuti 1 Definizioni e affermazione 2 Applicazioni 3 Prove 4 Dependence on choice 5 generalizzazioni 6 Appunti 7 References Definitions and statement 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. The four-color theorem states that every finite graph that can be drawn without crossings in the Euclidean plane needs at most four colors; però, some graphs with more complicated connectivity require more than four colors. 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. Perciò, 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 (ancora, assuming the axiom of choice) they can be calculated from the chromatic numbers of their finite subgraphs. Lo afferma, if the chromatic numbers of the finite subgraphs of a graph {stile di visualizzazione G} have a finite maximum value {stile di visualizzazione c} , then the chromatic number of {stile di visualizzazione G} itself is exactly {stile di visualizzazione c} . D'altro canto, if there is no finite upper bound on the chromatic numbers of the finite subgraphs of {stile di visualizzazione G} , then the chromatic number of {stile di visualizzazione G} itself must be infinite. Applications 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 {stile di visualizzazione k} , it also has a {stile di visualizzazione (2k+1)} -coloring. For finite graphs this follows because such graphs always have a vertex of degree at most {displaystyle 2k} , which can be colored with one of {displaystyle 2k+1} colors after all the remaining vertices are colored recursively. Infinite graphs with such an orientation do not always have a low-degree vertex (per esempio, Bethe lattices have {displaystyle k=1} but arbitrarily large minimum degree), so this argument requires the graph to be finite. But the De Bruijn–Erdős theorem shows that a {stile di visualizzazione (2k+1)} -coloring exists even for infinite graphs. A seven-coloring of the plane, and the four-chromatic Moser spindle drawn as a unit distance graph in the plane, providing upper and lower bounds for the Hadwiger–Nelson problem.

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. Perciò, the chromatic number of the plane must be either 5, 6, o 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 {displaystyle w} if and only if it has a partition into {displaystyle w} chains. 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 that every graph that can be drawn without crossings in the plane, finito o infinito, can be colored with four colors. Più generalmente, every infinite graph for which all finite subgraphs are planar can again be four-colored. Proofs The original proof of the De Bruijn–Erdős theorem, by De Bruijn, used transfinite induction. Gottschalk (1951) provided the following very short proof, based on Tychonoff's compactness theorem in topology. Supporre che, for the given infinite graph {stile di visualizzazione G} , every finite subgraph is {stile di visualizzazione k} -colorable, e lascia {stile di visualizzazione X} be the space of all assignments of the {stile di visualizzazione k} colors to the vertices of {stile di visualizzazione G} (regardless of whether they form a valid coloring). Quindi {stile di visualizzazione X} may be given a topology as a product space {stile di visualizzazione k^{V(G)}} , dove {stile di visualizzazione V(G)} denotes the set of vertices of the graph. By Tychonoff's theorem this topological space is compact. For each finite subgraph {stile di visualizzazione F} di {stile di visualizzazione G} , permettere {stile di visualizzazione X_{F}} be the subset of {stile di visualizzazione X} consisting of assignments of colors that validly color {stile di visualizzazione F} . Then the system of sets {stile di visualizzazione X_{F}} is a family of closed sets with the finite intersection property, so by compactness it has a nonempty intersection. Every member of this intersection is a valid coloring of {stile di visualizzazione G} . A different proof using Zorn's lemma was given by Lajos Pósa, and also in the 1951 dottorato di ricerca. thesis of Gabriel Andrew Dirac. Se {stile di visualizzazione G} is an infinite graph in which every finite subgraph is {stile di visualizzazione k} -colorable, then by Zorn's lemma it is a subgraph of a maximal graph {stile di visualizzazione M} with the same property (one to which no more edges may be added without causing some finite subgraph to require more than {stile di visualizzazione k} colors). The binary relation of nonadjacency in {stile di visualizzazione M} is an equivalence relation, and its equivalence classes provide a {stile di visualizzazione k} -coloring of {stile di visualizzazione G} . Tuttavia, this proof is more difficult to generalize than the compactness proof. The theorem can also be proved using ultrafilters or non-standard analysis. Nash-Williams (1967) gives a proof for graphs with a countable number of vertices based on Kőnig's infinity lemma.

Dependence on choice 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. Più precisamente, Mycielski (1961) 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 Läuchli (1971) showed that the De Bruijn–Erdős theorem and the Boolean prime ideal theorem are equivalent in axiomatic power. The De Bruijn–Erdős theorem for countable graphs can also be shown to be equivalent in axiomatic power, within a theory of second-order arithmetic, to Kőnig's infinity lemma. For a counterexample to the theorem in models of set theory without choice, permettere {stile di visualizzazione G} be an infinite graph in which the vertices represent all possible real numbers. In {stile di visualizzazione G} , connect each two real numbers {stile di visualizzazione x} e {stile di visualizzazione y} by an edge whenever one of the values {stile di visualizzazione |x-y|pm {mq {2}}} is a rational number. Equivalentemente, in this graph, edges exist between all real numbers {stile di visualizzazione x} and all real numbers of the form {displaystyle x+qpm {mq {2}}} , for rational numbers {stile di visualizzazione q} . Each path in this graph, starting from any real number {stile di visualizzazione x} , alternates between numbers that differ from {stile di visualizzazione x} by a rational number plus an even multiple of {stile di visualizzazione {mq {2}}} and numbers that differ from {stile di visualizzazione x} by a rational number plus an odd multiple of {stile di visualizzazione {mq {2}}} . This alternation prevents {stile di visualizzazione G} from containing any cycles of odd length, so each of its finite subgraphs requires only two colors. Tuttavia, in the Solovay model in which every set of real numbers is Lebesgue measurable, {stile di visualizzazione G} requires infinitely many colors, since in this case each color class must be a measurable set and it can be shown that every measurable set of real numbers with no edges in {stile di visualizzazione G} must have measure zero. Perciò, in the Solovay model, il (infinito) chromatic number of all of {stile di visualizzazione G} is much larger than the chromatic number of its finite subgraphs (at most two). Generalizations Rado (1949) proves the following theorem, which may be seen as a generalization of the De Bruijn–Erdős theorem. Permettere {stile di visualizzazione V} be an infinite set, for instance the set of vertices in an infinite graph. For each element {stile di visualizzazione v} di {stile di visualizzazione V} , permettere {stile di visualizzazione c_{v}} be a finite set of colors. Inoltre, for every finite subset {stile di visualizzazione S} di {stile di visualizzazione V} , choose some particular coloring {stile di visualizzazione C_{S}} di {stile di visualizzazione S} , in which the color of each element {stile di visualizzazione v} di {stile di visualizzazione S} belongs to {stile di visualizzazione c_{v}} . Then there exists a global coloring {stile di visualizzazione chi } of all of {stile di visualizzazione V} with the property that every finite set {stile di visualizzazione S} has a finite superset {stile di visualizzazione T} on which {stile di visualizzazione chi } e {stile di visualizzazione C_{T}} agree. In particolare, if we choose a {stile di visualizzazione k} -coloring for every finite subgraph of an infinite graph {stile di visualizzazione G} , then there is a {stile di visualizzazione k} -coloring of {stile di visualizzazione G} in which each finite graph has a larger supergraph whose coloring agrees with the coloring of the whole graph. 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. Per esempio, unboundedly chromatic graphs must also contain every possible finite bipartite graph as a subgraph. Tuttavia, they may have arbitrarily large odd girth, and therefore they may avoid any finite set of non-bipartite subgraphs. 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 {stile di visualizzazione k} -coloring if and only if each of its finite sub-hypergraphs has a {stile di visualizzazione k} -coloring. It is a special case of the compactness theorem of Kurt Gödel, stating that a set of first-order sentences has a model if and only if every finite subset of it has a model. Più specificamente, the De Bruijn–Erdős theorem can be interpreted as the compactness of the first-order structures whose non-logical values are any finite set of colors and whose only predicate on these values is inequality. The theorem may also be generalized to situations in which the number of colors is an infinite cardinal number. Se {displaystyle kappa } is a strongly compact cardinal, then for every graph {stile di visualizzazione G} and cardinal number {displaystyle lui 3.0.CO;2-e, SIG 1791549. Shela, Sahara; Soifer, Alessandro (2003), "Axiom of choice and chromatic number of the plane", Rivista di teoria combinatoria, Serie A, 103 (2): 387–391, doi:10.1016/S0097-3165(03)00102-X, SIG 1996076. Soifer, Alessandro (2008), The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, New York: Springer, ISBN 978-0-387-74640-1. See especially Chapter II.5 "De Bruin–Erdős reduction to finite sets and results near the lower bound", pp. 39–42, and Chapter V.26 "De Bruin–Erdős's theorem and its history", pp. 236–241. Categorie: Graph coloringInfinite graphsTheorems in graph theoryAxiom of choicePaul Erdős

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