论文标题
概率的Hadwiger-Nelson问题的有限图方法
A Finite Graph Approach to the Probabilistic Hadwiger-Nelson Problem
论文作者
论文摘要
我们推进了最初由PolyMath16项目开发的Hadwiger-Nelson问题的概率方法，特别是将有限单位距离图的方法联系起来。我们将平面的$ k $ - 颜色的数值\ textit {textit {dadments}定义是在着色下随机选择的单位距离边缘是单色的，并且我们使用与有限图形有关的概率技术的任意$ k $ - 颜色的不良界限。最终的界限的构型使我们能够根据非$ k $可加油的单位距离图计算下限，从而改善了Pritikin产生的界限和Polymath16项目在$ k = 4 $和$ k = 5 $的情况下。此外，我们在de bruijn-erds compactness定理的概率类似物上做出了部分进步。
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-distance graphs, improving bounds produced by Pritikin and the Polymath16 project in the $k = 4$ and $k = 5$ cases. Additionally, we make partial progress on a probabilistic analog of the de Bruijn-Erdős compactness theorem.
