论文标题
用六角形铺平飞机：$ k $颜色的改进的分离
Tiling the plane with hexagons: improved separations for $k$-colourings
论文作者
论文摘要
自1950年以来，人们就一直在众所周知，可以将七种颜色分配给具有单位直径细胞的无限蜂窝的瓷砖，因此，没有相同颜色的两个瓷砖比$ d（7）= \ frac {\ frac {\ sqrt {7}}} {2} $分开。各种作者都使用$ k> 7 $颜色描述了瓷砖，给出了$ d（k）$的相应值，但是通常未知这些是否是给定$ k $的最大值。在这里，对于许多$ k $，我们描述的是$ d（k）$的瓷砖，比以前报道的更大。
It has been common knowledge since 1950 that seven colours can be assigned to tiles of an infinite honeycomb with cells of unit diameter such that no two tiles of the same colour are closer than $d(7)=\frac{\sqrt{7}}{2}$ apart. Various authors have described tilings using $k>7$ colours, giving corresponding values for $d(k)$, but it is generally unknown whether these are the largest possible for a given $k$. Here, for many $k$, we describe tilings with larger values of $d(k)$ than previously reported.
