Four color theorem for 'solid' maps Is there an equivalent of the four color theorem for 'solid' maps? In other words, if we consider a 'map' in $3D$ what is the minimum number of colors we have to use in order to avoid that two adiacent solid regions have the same color?
Thanks.
 A: André's comment is correct: for any integer $n>0$ it is possible to produce a collection of $n$ solid regions each of which touches all the others. The solid regions to be constructed are best explained by a picture, but I don't want to put in the effort, so here's a description. The $n$ regions, $R_i$, for $0<i<n$ are all constructed in a similar way:


*

*Take two sticks each with unit square cross-sections and length $n$ units.

*Lay one stick perpendicularly on top of the other, so they form a cross.

*The lower stick will project $i$ units in front of the crossing stick and the crossing stick similarly projects $i$ units to the left of the lower stick.


Now take the $n$ regions and lay them out in left-to-right order $R_0, R_1, \dotsc, R_{n-1}$, arranged so that the lower stick of each is aligned in the front.
You'll then have $n$ regions, each of which touches all the rest. In short, there's no minimum chromatic number for solid regions in 3-space. An interesting followup: what if we required the regions to be convex?.  
