图的非友好或弱非友好划分
Unfriendly or weakly unfriendly partitions of graphs
摘要：
对于每个无限基数$κ$和每个图$G=(V,E)$，我们说一个划分$π:V\rightarrow \left\{ 0,1\right\} $是$κ$ -不友好的，如果对于每个$x\in V$，$\left| \left\{ y\in V\mid \left\{ x,y\right\} \in E\text{ and }π(y)\neq π(x)\right\} \right| $是$\geq \left| \left\{ y\in V\mid \left\{ x,y\right\} \in E\text{ and }π(y)=π(x)\right\} \right| $或$\geq κ$ ; $π$是不友好的，如果第一个属性对每个$x\in V$为真。S. Shelah和E.C.米尔纳，但目前还不知道，如果可数图没有不友好的分区存在。我们证明了，对于每个整数$n$，每个图的无限小度有一个$ω_{n}$ -不友好的分区。我们还证明了以下性质是等价的：（i）每个图都有$ω$ -不友好划分;（ii）每个可数图都有不友好划分;（iii）每个不含无限最小度的非空导出子图的可数图都有不友好划分（实际上考虑更小的一类图就足够了）。
英文摘要：
For each infinite cardinal $κ$ and each graph $G=(V,E)$, we say that a partition $π:V\rightarrow \left\{ 0,1\right\} $ is $κ$-unfriendly if, for each $x\in V$, $\left| \left\{ y\in V\mid \left\{ x,y\right\} \in E\text{ and }π(y)\neq π(x)\right\} \right| $ is $\geq \left| \left\{ y\in V\mid \left\{ x,y\right\} \in E\text{ and }π(y)=π(x)\right\} \right| $ or $\geq κ$; $π$ is unfriendly if the first property is true for each $x\in V$. Some uncountable graphs of infinite minimum degree without unfriendly partition have been constructed by S. Shelah and E.C. Milner, but it is not presently known if countable graphs without unfriendly partition exist. We show that, for each integer $n$, each graph of infinite minimum degree has an $ω_{n}$-unfriendly partition. We also prove that the following properties are equivalent: (i) each graph has an $ω$-unfriendly partition; (ii) each countable graph has an unfriendly partition; (iii) each countable graph without nonempty induced subgraph of infinite minimum degree has an unfriendly partition (actually it is enough to consider a smaller class of graphs).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
