Monochromatic Infinite Sets in Minkowski Planes

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