Recall that chromatic number of $\mathbb{R}^2$ is the least $n$ such that there exists a function $f$ from $\mathbb{R}^2$ into a set of colors ${C\_1,\ldots,C\_n}$ with $f(x)\neq f(y)$ for $||x-y||\_2=1$.


As far as I know, the problem which number this exactly is is still open. I was wondering whether this number is invariant under the norm $||\cdot||$ that is chosen.



Is the chromatic number of the real plane invariant under the norm?
For the $L^\infty$ norm (or equivalently the $L^1$ norm rotating by 45 degrees) the chromatic number is easy to calculate: 4. Just colour translates of $[0,1)\times[0,1)$ by $\mathbb Z^2$ in a $2\times 2$ repeating pattern. It must be at least 4 because ${0,1}\times{0,1}$ are all distance 1 apart.


There's certainly no simple periodic colouring with 4 colours that will work for the $L^2$ norm.


