论文标题
哈林的终端猜想
Halin's end degree conjecture
论文作者
论文摘要
图$ g $的结尾是射线的等效类别，如果它们之间有无限的顶点 - 偶口路径在$ g $中，则两个射线是等效的。末端的程度是该等价类中成对分离射线集合的最大基数。 哈林（Halin）猜想，可以通过某些典型的射线配置来表征末端度，这将概括其著名的\ emph {grid theorem}。 In particular, every end of regular uncountable degree $κ$ would contain a \emph{star of rays}, i.e.\ a configuration consisting of a central ray $R$ and $κ$ neighbouring rays $(R_i \colon i < κ)$ all disjoint from each other and each $R_i$ sending a family of infinitely many disjoint paths to $R$ so that paths from distinct families only meet in $ r $。 我们表明，Halin的猜想因最终程度$ \ aleph_1 $而失败，以$ \ aleph_2，\ aleph_3，\ ldots，\aleph_Ω$持有，$ \ aleph_ {ω+1} $失败了，在下一个$ \ aleph_ aleph_ al}中是$ \ n} $+n} $+n} \ Mathbb {n} $，$ n \ geq 2 $。进一步的结果包括针对GCH下所有红衣主教的完整解决方案，并取决于许多一致性结果。
An end of a graph $G$ is an equivalence class of rays, where two rays are equivalent if there are infinitely many vertex-disjoint paths between them in $G$. The degree of an end is the maximum cardinality of a collection of pairwise disjoint rays in this equivalence class. Halin conjectured that the end degree can be characterised in terms of certain typical ray configurations, which would generalise his famous \emph{grid theorem}. In particular, every end of regular uncountable degree $κ$ would contain a \emph{star of rays}, i.e.\ a configuration consisting of a central ray $R$ and $κ$ neighbouring rays $(R_i \colon i < κ)$ all disjoint from each other and each $R_i$ sending a family of infinitely many disjoint paths to $R$ so that paths from distinct families only meet in $R$. We show that Halin's conjecture fails for end degree $ \aleph_1$, holds for $\aleph_2,\aleph_3,\ldots,\aleph_ω$, fails for $ \aleph_{ω+1}$, and is undecidable (in ZFC) for the next $\aleph_{ω+n}$ with $n \in \mathbb{N}$, $n \geq 2$. Further results include a complete solution for all cardinals under GCH, complemented by a number of consistency results.
