Editing Four color theorem (section)

===Infinite graphs===
[[File:Torus with seven colours.svg|thumb|300px|By joining the single arrows together and the double arrows together, one obtains a [[torus]] with seven mutually touching regions; therefore seven colors are necessary.]]
[[File:Projection color torus.png|480px|thumb|This construction shows the torus divided into the maximum of seven regions, each one of which touches every other.]]
The four color theorem applies not only to finite planar graphs, but also to [[infinite graph]]s that can be drawn without crossings in the plane, and even more generally to infinite graphs (possibly with an uncountable number of vertices) for which every finite subgraph is planar. To prove this, one can combine a proof of the theorem for finite planar graphs with the [[De Bruijn–Erdős theorem (graph theory)|De Bruijn–Erdős theorem]] stating that, if every finite subgraph of an infinite graph is ''k''-colorable, then the whole graph is also ''k''-colorable {{harvtxt|Nash-Williams|1967}}. This can also be seen as an immediate consequence of [[Kurt Gödel]]'s [[compactness theorem]] for [[first-order logic]], simply by expressing the colorability of an infinite graph with a set of logical formulae.