Monochromatic infinite sets in Minkowski planes

We prove that for any $\ell_p$-norm in the plane with $1<p<\infty$ 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 $\mathcal{M}$.
Comment: 9 pages, 2 figures