Monochromatic infinite sets in Minkowski planes

TL;DR

This paper investigates monochromatic infinite sets in Minkowski planes, showing that for certain norms like $\, ext{l}_p$ with 1<p<∞, such sets can be avoided, while for polygonal norms they cannot.

Contribution

It establishes a dichotomy in Minkowski planes: avoidance of monochromatic infinite sets for $\, ext{l}_p$ norms and unavoidable sets for polygonal norms.

Findings

Avoidance of monochromatic sets in $\, ext{l}_p$-norms with 1<p<∞.

Existence of unavoidable monochromatic sets in polygonal norms.

Demonstrates a clear contrast between different Minkowski plane norms.

Abstract

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 $\mathcal{M}$.