Abstract—
Given \(1 \leqslant p \leqslant \infty \) and \(n \in \mathbb{N}\), we construct a two-coloring of the n-dimensional space \(\mathbb{R}_{p}^{n}\) equipped with the \({{\ell }_{p}}\) norm such that all sufficiently long unit arithmetic progressions contain points of both colors.
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Funding
This work was supported in part by the Program “Leading Scientific Schools” (grant no. NSh-775.2022.1.1) and by the Russian Foundation for Basic Research (project no. 20-31-90009). Sagdeev is a winner of the contest “Young Russian Mathematics” and is grateful to its sponsors and referees.
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Kirova, V.O., Sagdeev, A.A. Two-Colorings of Normed Spaces with No Long Monochromatic Unit Arithmetic Progressions. Dokl. Math. 106, 348–350 (2022). https://doi.org/10.1134/S1064562422050143
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DOI: https://doi.org/10.1134/S1064562422050143