一个概率Hadwiger-Nelson问题

某些域上的Hadwiger-Nelson问题

The Hadwiger-Nelson problem over certain fields

我们计算Hadwiger-Nelson数$χ(E^2)$为某些数字字段$E$，也就是说，颜色所需的最小数量的点在平面上的坐标在~ $E$，使没有两个点在距离$1$有相同的颜色。具体来说，我们表明，$χ(\mathbb{Q}(\sqrt{2})^2) = 2$，$χ(\mathbb{Q}(\sqrt{3})^2) = 3$，$χ(\mathbb{Q}(\sqrt{7})^2) = 3$尽管事实上，图$Γ(\mathbb{Q}(\sqrt{7})^2)$是三角形自由，和$4 \leq χ(\mathbb{Q}(\sqrt{3}, \sqrt{11})^2) \leq 5$。我们还讨论了其他域上的一些结果，对于其他二次域。最后，我们对选择公理的使用作了一些评论。

Probabilistic formulation of the Hadwiger--Nelson problem

This is a slightly modified version of my MSc thesis from 2019. Supervisor was Dömötör Pálvölgyi.

A Finite Graph Approach to the Probabilistic Hadwiger-Nelson Problem

We advance a probabilistic approach to the Hadwiger-Nelson problem initially developed by the Polymath16 project, in particular relating the approach to finite unit-distance graphs. We define the numerical \textit{badness} of a given $k$-coloring of the plane to be the probability that a randomly chosen unit-distance edge is monochromatic under the coloring, and we provide lower bounds on the badness of arbitrary $k$-colorings using a probabilistic technique relating to finite graphs. The contrapositive of the resulting bounds lets us compute lower bounds on the order of non $k$-colorable unit…

The Hadwiger-Nelson problem with two forbidden distances

In 1950 Edward Nelson asked the following simple-sounding question: \emph{How many colors are needed to color the Euclidean plane $\mathbb{E}^2$ such that no two points distance $1$ apart are identically colored?} We say that $1$ is a \emph{forbidden} distance. For many years, we only knew that the answer was $4$, $5$, $6$, or $7$. In a recent breakthrough, de Grey \cite{degrey} proved that at least five colors are necessary. In this paper we consider a related problem in which we require \emph{two} forbidden distances, $1$ and $d$. In other words, for a given positive number $d\neq 1$, how ma…

一类由Hadwiger Nelson问题产生的新的几何定义超图

A New Class of Geometrically Defined Hypergraphs Arising from the Hadwiger Nelson Problem

针对欧几里得空间单位距离图色数问题，通过推广Erdos-De-Bruijn定理构造有限图序列，得到一类几何定义超图，其正常着色与单位距离图一致，并部分推广至任意实赋范向量空间。

Hadwiger-Nelson问题的一个推广

On a generalization of the Hadwiger-Nelson problem

对于域$F$和定义在$F$上的$n$维向量空间$V$上的二次型$Q$，令$\mathrm{QG}_Q$，称为与$Q$相关联的二次图，是具有顶点集$V$的图，其中顶点$u,w \in V$形成边当且仅当$Q(v-w)=1$。二次图可以被看作是著名的Hadwiger-Nelson问题中的单位距离图的自然推广。本文证明了对特征为零的局部域$F$，$\mathrm{QG}_Q$的Borel色数是无穷的当且仅当$Q$在$F$上非平凡地表示零.证明采用了最近的频谱界的凯莱图的Borel色数，结合局部领域的某些振荡积分的分析。作为应用，我们还将回答2009年第22届英国组合数学会议上提出的问题525的一个变体。