Menger's theorem is an important building block of numerous results in the study of graph structure. We consider a variant in terms of coarse geometry. We say that a set of graphs has the weak coarse Menger property if there exist functions $f$ and $g$ such that for any graph $G$ in this set, subsets $X$ and $Y$ of vertices of $G$, and positive integers $k$ and $r$, either there exist $k$ paths between $X$ and $Y$ pairwise at distance at least $r$, or there exists a union of at most $f(k,r)$ balls of radius at most $g(k,r)$ intersecting all paths between $X$ and $Y$. Nguyen, Scott and Seymour proved that the set of all graphs does not have the weak coarse Menger property and asked whether every proper minor-closed family of finite graphs has it. In this paper, we provide a positive answer to this question in a stronger form: it is true for the set of locally finite graphs with an excluded finite minor, and the functions $f$ and $g$ can be chosen so that $f$ only depends on the number $k$ of the paths in the packing and the function $g$ is a linear function of the distance threshold $r$ and is independent of $k$, which is optimal up to a constant factor. Our result extends to every length space quasi-isometric to a locally finite graph or metric graph with an excluded finite minor, such as complete Riemannian surfaces of finite Euler genus, string graphs, and Cayley graphs of finitely generated minor-excluded groups.

翻译：Menger定理是图结构研究中众多结果的重要基石。本文从粗几何角度考虑该定理的一个变体。我们称一族图具有弱粗Menger性质，若存在函数$f$和$g$，使得对该族中任意图$G$、顶点子集$X$与$Y$、以及正整数$k$和$r$，要么存在$k条$两两距离至少为$r$的$X$- $Y$路径，要么存在至多$f(k,r)$个半径为$g(k,r)$的球的并集，与所有$X$- $Y$路径相交。Nguyen、Scott与Seymour证明了所有图构成的族不具有弱粗Menger性质，并询问每个真极小闭的有限图族是否具有该性质。本文给出该问题的肯定回答，且结论更强：对局部有限且包含有限极小结构的图族成立，并且函数$f$和$g$可选取为$f$仅依赖于路径包数$k$，而$g$是距离阈值$r$的线性函数且与$k$无关——该结果在常数因子意义下最优。该结论可推广至所有与局部有限图或度量图拟等距的长度空间（这些空间包含有限极小结构），例如有限欧拉亏格的完备黎曼曲面、弦图、以及有限生成极小除外群的Cayley图。