Four Color Map Theorem In Higher Dimensions?

[Bohandas]

Is there an equivalent of the four-color map theorem for higher dimensions?

For example, given a bunch of three dimensional objects that are touching each other is there a maximum number of colors that would be required so that no object would be the same color as another object that it's touching along part of a face?

(I know it works in fewer dimensions, it's clear that in one dimension you need only two, (unless the line is circular and joins itself in which case you need three))

[factotum]

The article on Wikipedia about the four colour theorem says this:

"There is no obvious extension of the coloring result to three-dimensional solid regions. By using a set of n flexible rods, one can arrange that every rod touches every other rod. The set would then require n colors, or n+1 if you consider the empty space that also touches every rod. The number n can be taken to be any integer, as large as desired."

If it's not possible in three dimensions I think it vanishingly unlikely it will be in four, five or what-have-you.

[Fat Rooster]

No. Complete graphs are easily constructed in any more than two dimensions. If we draw points on a 2 dimensional surface then forming connections between them will always form a boundary that interferes with adding more connections. In 3 dimensions we can simply go over or under those connections, and they do not interfere with each other.

[Jay R]

The closest equivalent is that surfaces of some three-dimensional objects have a different number. A map on the surface of a torus (doughnut shape) can require up to seven colors, for instance.