may be seen as a generalization of the De Bruijn–Erdős theorem. Let be an infinite set, for instance the set of vertices in an infinite graph. For each element of , let be a finite set of colors. Additionally, for every finite subset of , choose some particular coloring of , in which the color of each element of belongs to . Then there exists a global coloring of all of with the property that every finite set has a finite superset on which and agree. In particular, if we choose a -coloring for every finite subgraph of an infinite graph , then there is a -coloring of 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. 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. 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 -coloring if and only if each of its finite sub-hypergraphs has a -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. More specifically, 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. If is a strongly compact cardinal, then for every graph and cardinal number , has chromatic number at most if and only if each of its subgraphs of cardinality less than has chromatic number at most . The original De Bruijn–Erdős theorem is the case of this generalization, since a set is finite if and only if its cardinality is less than .