Four colour problem
It’s hard to prove mathematically, but it’s not very hard to wrap your brain around. In 2 dimensional space, it is impossible to draw five shapes that all touch each other with a contiguous border. Try it. Once you have four or more shapes, making the four touch requires that one shape be fully enveloped. You could have one shape touch as many shapes as you like, but those shapes cannot be touching more than three of the other shapes.
Are their exotic 2D geometries that work? Like, for example, the surface of a Klein bottle? (A cursory google search rules out the Klein bottle – any others perhaps?)
Not for contiguous spaces. Think of a triangle surrounded by three rectangles. There’s no way for all three outer rectangles to touch without enveloping the center triangle. There are only two directions you can go, and as long as the center triangle is enveloped, it cannot connect to a fifth shape without separating two of the outer shapes.
Alright, non Euclidean maps it is then!
On a torus, you can have up to seven mutually adjacent regions. See https://upload.wikimedia.org/wikipedia/commons/3/37/Projection_color_torus.png
Oh, very nice!
