The Hadwiger-Nelson problem over certain fields

Abstract

We compute the Hadwiger-Nelson numbers $\chi (E^2)$ for certain number fields $E$, that is, the smallest number of colors required to color the points in the plane with coordinates in~$E$ so that no two points at distance $1$ from one another have the same color. Specifically, we show that $\chi (\mathbb{Q}(\sqrt{2}{)}^2)=2$, that $\chi (\mathbb{Q}(\sqrt{3}{)}^2)=3$, that $\chi (\mathbb{Q}(\sqrt{7}{)}^2)=3$ despite the fact that the graph $\Gamma (\mathbb{Q}(\sqrt{7}{)}^2)$ is triangle-free, and that $4\le \chi (\mathbb{Q}(\sqrt{3},\sqrt{11}{)}^2)\le 5$. We also discuss some results over other fields, for other quadratic fields. We conclude with some comments on the use of the axiom of choice.

preliminary version

Cited by in corpus (1)

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