Monochromatic infinite sets in Minkowski planes

We prove that for any ${\mathrm{\ell }}_p$-norm in the plane with $1<p<\mathrm{\infty }$ and for every infinite $\mathcal{M}\subset {\mathbb{R}}^2$, there exists a two-colouring of the plane such that no isometric copy of $\mathcal{M}$ is monochromatic. On the contrary, we show that for every polygonal norm (that is, the unit ball is a polygon) in the plane, there exists an infinite $\mathcal{M}\subset {\mathbb{R}}^2$ such that for every two-colouring of the plane there exists a monochromatic isometric copy of $\mathcal{M}$.