无长单色单位等差列赋范空间的双着色

SIAM J. Discret. Math. Pub Date : 2022-03-09 DOI:10.1137/22M1483700

V. Kirova, A. Sagdeev

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引用次数: 2

摘要

给定一个自然的$n$，我们构造一个双着色的$\mathbb{R}^n$，其最大度规满足以下条件。对于任何直径大于$5^{n}$的有限实数集$S$，且$S$的任意两个连续点之间的距离不超过1，则$S$的等距副本都不是单色的。作为推论，我们证明了任何赋范空间都可以是双色的，使得所有足够长的单位等差数列都包含两种颜色的点。

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Two-Colorings of Normed Spaces without Long Monochromatic Unit Arithmetic Progressions

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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SIAM J. Discret. Math.

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