图的笛卡尔幂的顶点-拉姆齐定理

中文摘要 AI 辅助

对于图G、H以及正整数r和n，若对G的笛卡尔幂G^□ⁿ进行任意r顶点着色，都包含单色的H副本，则记为G^□ⁿ → H。由于G^□ⁿ的色数χ与χ(G)相同，当r=χ(G)时，存在一种对G^□ⁿ的r顶点着色，使得每个颜色类都是独立集。我们证明，当r<χ(G)时，存在一大类图H满足G^□ⁿ → H，这类图是超立方体中的所谓分层图。我们还表明，对于某些图G，例如奇环或团，当χ(G)/2 < r < χ(G)时，分层图H类是唯一满足上述拉姆齐性质的图类。此外，我们证明了一个更具一般性的结果，关联G和H的拉姆齐性质，使得G^□ⁿ → H成立。技术工具之一是离散立方体[m]^n的拉姆齐型命题，我们称之为立方体分层引理，它本身具有独立研究价值。研究G的笛卡尔幂的拉姆齐性质的原始动机之一是：若G是单位距离图，则G^□ⁿ也是单位距离图，这在欧氏拉姆齐理论中具有应用。

英文摘要

For graphs $G,H$ and positive integers $r$ and $n$ we write $G^{\square n} \xrightarrow{r} H$ if every $r$-vertex-coloring of the Cartesian power $G^{\square n}$ of $G$ contains a monochromatic copy of $H$. Since chromatic number $χ$ of $G^{\square n}$ is the same as $χ(G)$, there is an $r$-vertex coloring of $G^{\square n}$ for $r=χ(G)$, such that each color class is an independent set. We prove that for $r<χ(G)$ there is a large class of graphs $H$ such that $G^{\square n} \xrightarrow{r} H$. These graphs are so-called layered graphs in a hypercube. We also show that for some graphs $G$, such as for example odd cycles or cliques, the class of layered graphs $H$ is the only one satisfying the above Ramsey property when $χ(G)/2 < r < χ(G)$. In addition, we prove a more general result relating Ramsey properties of $G$ and graphs $H$ such that $G^{\square n} \xrightarrow{r} H$. One of the technical tools is a Ramsey-type statement for discrete cubes $[m]^n$ that we call the Cube Layered Lemma, which is of independent interest. One of the original motivations for studying Ramsey properties of Cartesian powers of $G$ is the fact that $G^{\square n}$ is a unit distance graph if $G$ is a unit distance graph. This provides applications in Euclidean Ramsey theory.

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16 pages