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arXiv · 2203.04555

Two-colorings of normed spaces without long monochromatic unit arithmetic progressions

Valeriya Kirova·Arsenii Sagdeev
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Abstract

Given a natural $n$, we construct a two-coloring of $\mathbb{R}^n$ with the maximum metric satisfying the following. For any finite set of reals $S$ with diameter greater than $5^{n}$ such that the distance between any two consecutive points of $S$ does not exceed one, no isometric copy of $S$ is monochromatic. As a corollary, we prove that any normed space can be two-colored such that all sufficiently long unit arithmetic progressions contain points of both colors.

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Valeriya Kirova, Arsenii Sagdeev. 2022-03-09. Two-colorings of normed spaces without long monochromatic unit arithmetic progressions. https://doi.org/10.1137/22m1483700

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