典型范数的Hadwiger--Nelson问题
Hadwiger--Nelson Problem for Typical Norms
- Princeton University（普林斯顿大学）
- Tel Aviv University（特拉维夫大学）
- University of Vienna（维也纳大学）
- Leipzig University（莱比锡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究典型范数的Hadwiger--Nelson问题，将单位距离图色数的上界从指数改进为线性（至多2d且紧），并借助高维矩阵推广的Lonely runner猜想解决了相关猜想。
AI中文摘要：
经典的Hadwiger--Nelson问题询问欧几里得平面单位距离图的色数。多年来，这一问题已在多种其他赋范空间中考虑，其中高维欧几里得空间$\mathbb{R}^d$或许是最自然且被研究最多的。Alon、Bucić和Sauermann证明了，对于$\mathbb{R}^d$上的典型范数，单位距离图的色数至多为$2^d$。我们将这一指数界改进为线性界，证明$\mathbb{R}^d$上典型范数的色数至多为$2d$，且这一界是紧的。这表明与欧几里得情形相比行为存在显著差异，后者具有指数下界。关键要素之一是Lonely runner猜想的一个高维矩阵推广，这还使我们能够完全解决Schoenberg于1978年提出的所谓视障猜想以及Henze和Malikiosis提出的更近的覆盖半径猜想。
英文摘要：
The classical Hadwiger-Nelson problem asks for the chromatic number of the unit distance graph of the Euclidean plane. Over the years, this problem has been considered for a variety of other normed spaces, with higher-dimensional Euclidean space $\mathbb{R}^d$ being perhaps the most natural and well-studied. Alon, Bucić, and Sauermann proved that, for a typical norm on $\mathbb{R}^d$, the chromatic number of the unit distance graph is at most $2^d$. We improve this exponential bound to a linear one by showing that the chromatic number of a typical norm on $\mathbb{R}^d$ is at most $2d$ and that this is tight. This shows a stark difference in the behavior compared to the Euclidean case, where there is an exponential lower bound. One of the key ingredients is a certain high-dimensional, matrix generalization of the Lonely runner conjecture, which also allows us to completely settle the so-called view-obstruction conjecture of Schoenberg from 1978 and a more recent covering-radius conjecture of Henze and Malikiosis.
