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I am thinking on the analogy of the well-known 2D coloring problem for the 3D space (with the trivial geometry & topology).

As this reference says, simply increasing the dimensions by one doesn't work. It would elevate the color number to infinite, because:

In this case, however, once you go to three dimensions, you can make partitions of space into regions for which you need N colors to color the regions in order that no two adjacent regions will have the same color for any N.

You can make an example by starting with one ball.

Now, add a ball to the picture and connect it with a thin tube to the first ball.

Now, add a third ball to the picture and connect this ball with two thin tubes to the two balls already in the picture.

You can keep adding balls and connecting them to all the other balls like this because there is enough space in three dimensions to work with.

If the balls represent regions, since each ball is touching every other ball, you need at least as many colors as there are balls to color them.

I think it would be useful if we would add a restriction: all of the regions should be convex. Also I think not I am the first one who thought on this possibility.

Is it possible? How to even start to think on such a problem?

What could be the result?

Glorfindel
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peterh
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  • I guess the result with convex bodies is still infinite, though it would take some more fiddling around to achieve. – Ivan Neretin Feb 01 '17 at 13:13
  • @IvanNeretin Thanks. I've fiddled a lot around it and I couldn't construct a really infinite one. I hit mainly similar walls as I tried naively to construct not 4-colorable 2d regions. – peterh Feb 01 '17 at 13:26

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I actually considered this very problem some years ago and managed to convince myself that the answer should be finite if we restricted to convex regions (I thought it would be less than 20).

Unfortunately, it seems that even when we restrict to convex regions, the 'colour number' can still be arbitrarily large. There are two references which I've found online which claim to prove this, specifically when we restrict to just using cuboid regions.

Painting the Office by Reed and Allwright.
Colouring Rectangular Blocks in 3-space by Magnant and Martin

The first paper also has references to earlier works where it was proved that the colour number is unbounded for convex regions.

hexomino
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