不可数有向图中的大型顶点火焰
Large vertex-flames in uncountable digraphs
AI总结：
本文将 Lovász 关于有限有向图中大型火焰的最优性结构刻画推广到大小为 ℵ₁ 的不可数有向图，克服了远超可数情形的技术困难，推进了对任意基数情形的理解。
摘要：
对见证连通性性质的最小子图的研究是图论中的一个重要领域。大型火焰的研究基础由 Lovász 奠定：设 $D=(V,E)$ 是一个有限有向图，$r\in V$。从 $r$ 到 $v$ 的局部连通性 $κ_D(r,v)$ 定义为 $D$ 中内部不相交的 $r\rightarrow v$ 路径的最大数目。若 $D$ 的生成子有向图 $L$ 对每个 $v\in V-r$ 满足 $κ_L(r,v)=κ_D(r,v)$，则它必须至少有 $\sum_{v\in V-r}κ_D(r,v)$ 条边。Lovász 证明了，也许令人惊讶的是，这个下界对每个有限有向图都是精确的。满足 Lovász 定理中最小-最大判据的 $L$ 的最优性也可以通过以下结构刻画来把握：对每个 $v\in V-r$，在 $L$ 中存在一个内部不相交的 $r\rightarrow v$ 路径系统 $\mathcal{P}_v$，覆盖 $L$ 中所有进入 $v$ 的边，使得可以从每个 $P\in \mathcal{P}_v$ 中选择一条边或一个内部顶点，使所得集合与 $D$ 中每条 $r\rightarrow v$ 路径相交。基于这一结构性无穷推广，第二作者得到了可数无穷有向图上的正结果。本文将其推广到大小为 $\aleph_1$ 的有向图，这需要显著更复杂的技术。尽管解决了最小的不可数情形，但对这一概念的完全理解以及可能对任意基数的证明似乎仍然遥远。
英文摘要：
The study of minimal subgraphs witnessing a connectivity property is an important field in graph theory. The foundation for large flames has been laid by Lovász: Let $ D=(V,E) $ be a finite digraph and let $ r\in V $. The local connectivity $ κ_D(r,v) $ from $ r $ to $ v $ is defined to be the maximal number of internally disjoint $ r\rightarrow v $ paths in $ D $. A spanning subdigraph $ L $ of $ D $ with $ κ_L(r,v)=κ_D(r,v) $ for every $ v\in V-r $ must have at least $\sum_{v\in V-r}κ_D(r,v) $ edges. Lovász proved that, maybe surprisingly, this lower bound is sharp for every finite digraph. The optimality of an $ L $ sufficing the min-max criteria from Lovász' theorem may instead also be captured by the following structural characterization: For every $ v\in V-r $ there is a system $ \mathcal{P}_v $ of internally disjoint $ r\rightarrow v $ paths in $ L $ covering all the ingoing edges of $ v $ in $ L $ such that one can choose from each $ P\in \mathcal{P}_v $ either an edge or an internal vertex in such a way that the resulting set meets every $ r\rightarrow v $ path of $ D $. The positive result for countably infinite digraphs based on this structural infinite generalisation were obtained by the second author. In this paper we extend this to digraphs of size $ \aleph_1 $ which requires significantly more complex techniques. Despite solving yet the smallest uncountable case, the complete understanding of the concept and potentially a proof for arbitrary cardinality still seems to be far.
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