无长单色单位等差列赋范空间的双着色

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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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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