states that, if the chromatic numbers of the finite subgraphs of a graph have a finite maximum value , then the chromatic number of itself is exactly . On the other hand, if there is no finite upper bound on the chromatic numbers of the finite subgraphs of , then the chromatic number of 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 , it also has a -coloring. For finite graphs this follows because such graphs always have a vertex of degree at most , which can be colored with one of colors after all the remaining vertices are colored recursively. Infinite graphs with such an orientation do not always have a low-degree vertex (for instance, Bethe lattices have but arbitrarily large minimum degree), so this argument requires the graph to be finite. But the De Bruijn–Erdős theorem shows that a -coloring exists even for infinite graphs. 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