论文标题
单色无限子图的密度II
Density of monochromatic infinite subgraphs II
论文作者
论文摘要
1967年，Gerencsér和Gyárfás证明了一个结果，被认为是图形 - 斑点理论的起点：在$ k_n $的每2彩中，都有$ \ lceil（2n+1）/3 \ rceil $ Vertices上的单色路径，这是最好的。从那以后，关于图形 - 斑点理论的论文已有数百篇论文，其中一些最重要的结果是由伯尔和埃尔德（Burr and Erd \ h）的一系列猜想所激发的，这些猜想是关于树木数量的，具有界限的最大程度的图形以及具有界限归化性的图形。 In 1993, Erd\H os and Galvin \cite{EG} began the investigation of a countably infinite analogue of the Gerencsér and Gyárfás result: What is the largest $d$ such that in every $2$-coloring of $K_\mathbb{N}$ there is a monochromatic infinite path with upper density at least $d$. Erd \ H OS和Galvin表明$ 2/3 \ LEQ D \ LEQ 8/9 $，经过一系列最近的改进，最终证明了$ d = {（12+ \ sqrt {8} {8}）}/{17} $。 本文开始了一项针对定量无限图形理论的系统研究，重点介绍了burr-erds的无限类似物。我们获得了一些类似于有限情况的结果，以及在有限情况下没有类似物的其他（意外）结果。
In 1967, Gerencsér and Gyárfás proved a result which is considered the starting point of graph-Ramsey theory: In every 2-coloring of $K_n$ there is a monochromatic path on $\lceil(2n+1)/3\rceil$ vertices, and this is best possible. There have since been hundreds of papers on graph-Ramsey theory with some of the most important results being motivated by a series of conjectures of Burr and Erd\H os regarding the Ramsey numbers of trees, graphs with bounded maximum degree, and graphs with bounded degeneracy. In 1993, Erd\H os and Galvin \cite{EG} began the investigation of a countably infinite analogue of the Gerencsér and Gyárfás result: What is the largest $d$ such that in every $2$-coloring of $K_\mathbb{N}$ there is a monochromatic infinite path with upper density at least $d$. Erd\H os and Galvin showed that $2/3\leq d\leq 8/9$, and after a series of recent improvements, it was finally shown that $d={(12+\sqrt{8})}/{17}$. This paper begins a systematic study of quantitative countably infinite graph-Ramsey theory, focusing on infinite analogues of the Burr-Erdős conjectures. We obtain some results which are analogous to what is known in finite case, and other (unexpected) results which have no analogue in the finite case.
