This is potentially another approach to this question. I put it as an update there, but perhaps it would be better to post it separately. If we color $\mathbb{Z}^k$ with the $\ell_\infty$ metric in fewer than $k+1$ colors, then there are always arbitrary long monochromatic paths. This follows from the classical results about the HEX game. It is known that this result is "equivalent" (both results can be easily deduced from each other) to the Brouwer fixed point theorem (see also a discussion here ). Now consider the $\mathbb{Z}^k$ with the $\ell_1$-metric and a similar question as before: can we color it in fewer than $k+1$ colors, so that there are no arbitrary long monochromatic $3$-paths (i.e. we are allowed to jump by 1 or by 2 or by 3 in the $\ell_1$-metric).
Question. Is this statement equivalent (in the above sense) to a fixed point theorem.
