正规生成树的最小障碍
Minimal obstructions for normal spanning trees
摘要：
Diestel和Leader刻画了连通图的特征，通过两类禁止子图可以得到一个正常的生成树.一类是Halin的$(\aleph_0,\aleph_1)$ -图：具有二分图$(\mathbb{N},B)$使得$B$不可数且$B$的每个顶点都有无限度的二部图.我们的主要结果是，在Martin公理和连续统假设失败的情况下，Diestel和Leader结果中的禁止的$(\aleph_0,\aleph_1)$ -图类可以被这样的图的一个单个实例替换。然而，在CH下，$(\aleph_0,\aleph_1)$ -图类包含次要的不可比元素，即二元型图和$\mathcal{U}$ -不可分图。假设CH，Diestel和领导人问是否每一个$(\aleph_0,\aleph_1)$ -图有一个$(\aleph_0,\aleph_1)$ -未成年人是不可分的或二进制类型，以及是否任何两个$\mathcal{U}$ -不可分的图形必然是对方的未成年人。对于这两个问题，我们构造的例子表明，答案是否定的。
英文摘要：
Diestel and Leader have characterised connected graphs that admit a normal spanning tree via two classes of forbidden minors. One class are Halin's $(\aleph_0,\aleph_1)$-graphs: bipartite graphs with bipartition $(\mathbb{N},B)$ such that $B$ is uncountable and every vertex of $B$ has infinite degree. Our main result is that under Martin's Axiom and the failure of the Continuum Hypothesis, the class of forbidden $(\aleph_0,\aleph_1)$-graphs in Diestel and Leader's result can be replaced by one single instance of such a graph. Under CH, however, the class of $(\aleph_0,\aleph_1)$-graphs contains minor-incomparable elements, namely graphs of binary type, and $\mathcal{U}$-indivisible graphs. Assuming CH, Diestel and Leader asked whether every $(\aleph_0,\aleph_1)$-graph has an $(\aleph_0,\aleph_1)$-minor that is either indivisible or of binary type, and whether any two $\mathcal{U}$-indivisible graphs are necessarily minors of each other. For both questions, we construct examples showing that the answer is in the negative.
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。
