Consider the graph whose vertices are the elements of $\mathbb{R}^n$ with an edge between $x$ and $y$ iff $d(x,y)=1$. There is an extensive literature on the chromatic numbers $\chi(\mathbb{R}^n)$ (meaning vertex colorings) but I did not see the following. It is obvious that these chromatic numbers are increasing with $n$, are they strictly increasing? i.e. is $\chi(\mathbb{R}^n)<\chi(\mathbb{R}^{n+1})$ for each $n$?
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1$\begingroup$ I would be surprised if this is known. For example, it is known that $\chi(\mathbb{R}^2) \le 7$ and $\chi(\mathbb{R}^3)\ge 6$, but I don't think any better bounds are known. That doesn't quite rule out the possibility that $\chi(\mathbb{R}^2 )< \chi(\mathbb{R}^3)$ could be known, but again, I would be surprised. See On the chromatic numbers of small-dimensional Euclidean spaces for some more info. $\endgroup$– Timothy ChowCommented Dec 9 at 16:25
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