Graph theory, colourability in 3-space. Can somebody explain colourability in $3$-space or share really good material that I can read? I understand the rules of $4$-colourability in planar universe: sections sharing the same border cannot have the same colour but corners can, e.g. a square containing four equal subsquares, the first and third quadrants can have the same colour while the second and fourth share the same colour. But I cannot find any information about $3$-space/dimensions. I am taking a course in general mathematics. Everything from ancient history to modern history, from all possible areas of mathematics; but unfortunately the book doesn't dive much into colourability in $3$-space.
 A: The answer to your question is that no finite number of colors is sufficient to color every three-dimensional "map".  For example, it's possible to have 100 different galactic empires, every pair of which share a common border in space, which means you would need at least 100 different colors to color this three-dimensional "map".
To picture this, imagine 100 different empires whose territory consists of a spherical blob in space together with 99 thin tendrils that stretch out and touch the other empires.
A: I don't think that k-colorability requires a space to be in. That is, the original problem was about coloring maps, but was then generalized to coloring any graph - planar or not. So the general problem of a proper k-coloring is assigning colors (usually just numbers) to the vertices of a graph such that no edge has endpoints of the same color.
For a cartographic map, the vertices are countries and the edges are borders. However, the problem is the same for graphs that could not represent cartographic maps, as they are not drawable in the plane.
