This is potentially another approach to <a href="http://mathoverflow.net/questions/52825/coloring-mathbb-zk"> this question</a>. 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 <a href="http://www.math.pitt.edu/~gartside/hex_Browuer.pdf"> HEX game.</a> 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 <a href="http://mathoverflow.net/questions/19272/applications-of-brouwers-fixed-point-theorem"> here </a>). 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).

<b> Question. </b> Is this statement equivalent (in the above sense) to a fixed point theorem.