Monochromatic Infinite Sets in Minkowski Planes

Frankl, Nóra and Gehér, Panna and Sagdeev, Arsenii and Tóth, Géza (2024) Monochromatic Infinite Sets in Minkowski Planes. DISCRETE AND COMPUTATIONAL GEOMETRY. ISSN 0179-5376

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Abstract

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

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