Coloring vertices of cube

Question

I recently encountered a brain-teaser, which, after being stripped from literary flourishes, is reduced to the following:

take a graph $G$ composed of vertices and edges of a unit cube in $\Bbb R^N$
to each vertex of $G$ assign one of $N$ distinct colors such that each vertex is adjacent to vertices of all possible colors ("each vertex sees $N$ colors")
describe all $N$ for which this procedure is possible.

I can show by hand that for $N=1,2,4$ this is possible and impossible for $N=3$; I do not have necessary knowledge to solve this problem in general.

I wonder if there are entry-level sources on graph theory that would allow me to tackle this problem. Any pointers are welcome, too. Please feel free to edit the question to add relevant tags.

@ajotatxe The wiki article on Ramsay's problem talks about colored edges, not colored vertices. Is there something that I miss there? — TZakrevskiy, Jan 30 '24 at 13:58

score 1 · Answer 1 · answered Jan 26 '24 at 16:37

Let $c $ be some color within the graph. Let $k $ be the number of vertex colored $c $. Let $V $ be the vertex of the cube. Consider the following sum:

$\sum_{v\in V} \#($vertex adjacent of color $c$).

Because of the imposed condition, this sum is equal to exactly the number of vertex in the cube: $2^N$.

On the other hand, each vertex of color $c $ is counted exactly $N $ times thus we have: $N \cdot k = 2^N \Rightarrow N$ divides $2^N$ thus $N $ must be a power of $2$.

You have shown that "N a power of 2" is a necessary condition. But is it sufficient? — Calvin Lin, Jan 26 '24 at 17:50

Coloring vertices of cube

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