Summary
LetQ 4 denote the graph, obtained from the rational points of the 4-space, by connecting two points iff their Euclidean distance is one. It has been known that its chromatic number is 4. We settle a problem of P. Johnson, showing that in every four-colouring of this graph, every colour class is every-where dense.
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Zaks, J. On four-colourings of the rational four-space. Aeq. Math. 37, 259–266 (1989). https://doi.org/10.1007/BF01836448
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DOI: https://doi.org/10.1007/BF01836448