In the Hadwiger-Nelson problem, any two points unit distance apart must have distinct colors. However, it is known that if we restrict the vertices to only rational numbers, the chromatic number is equal to 2 (exactly). I cannot find a proof/reference anywhere so if someone can briefly explain the intuition/outline of the result, that would be much appreciated.
2026-03-27 13:46:15.1774619175
Hadwiger-Nelson problem only on Q rationals
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It is Theorem $7$ in this PDF, and there is a proof of the result for the integer lattice and an indication of how to modify it for the rational lattice here.