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.