论文标题
没有长单色单元算术进程的规范空间的两个颜色
Two-colorings of normed spaces without long monochromatic unit arithmetic progressions
论文作者
论文摘要
给定天然$ n $，我们构建了两种颜色的$ \ mathbb {r}^n $，最大度量满足以下内容。对于任何有限的REALS $ S $，直径大于$ 5^{n} $，以使任何两个连续的$ s $之间的距离不超过一个，否$ s $的均值副本是单色的。作为推论，我们证明任何规范的空间都可以是两色，以使所有足够长的单位算术进程都包含两种颜色的点。
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.
