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arXiv · 2308.08840

Monochromatic infinite sets in Minkowski planes

Nóra Frankl·Panna Gehér·Arsenii Sagdeev·Géza Tóth
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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 $\mathcal{M}$.

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Nóra Frankl, Panna Gehér, Arsenii Sagdeev, Géza Tóth. 2024-11-17. Monochromatic infinite sets in Minkowski planes. https://doi.org/10.1007/s00454-024-00702-5

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