发表机构
Leipzig University; Université de Bordeaux; CNRS（莱比锡大学; 波尔多大学; 法国国家科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
证明了粗埃尔德什-波萨猜想：无$k\cdot K_3$粗模型的图要么含许多相距远的粗环，要么被少量有界半径球覆盖，并进一步得到拟等距到无$k\cdot K_3$子式图的结果。
AI 中文摘要
我们证明了Georgakopoulos和Papasoglu提出的粗埃尔德什-波萨猜想。非正式地说，任何图要么包含许多两两相距很远的粗环，要么存在少量有界半径的球，这些球共同击中所有这些粗环。更精确地说，如果图$G$对于某些$q, k \in \mathbb{N}$没有$k \cdot K_3$的$q$-粗模型，那么存在一个大小为$\mathcal{O}(k\log k)$的顶点集$X\subseteq V(G)$，使得$G$中每个$K_3$的$q$-粗模型到$X$的距离为$\mathcal{O}(q)$。在另一种更接近刻画拟树的Manning定理的形式中：如果图$G$对于某些$q, k \in \mathbb{N}$没有$k \cdot K_3$的$q$-粗模型，那么$G$与一个图$H$是$\mathcal{O}(q)$-拟等距的，其中$H$包含一个大小为$\mathcal{O}(k\log k)$的顶点集$X\subseteq V(H)$，使得$H-X$是一个森林。通过自举这一结果，我们进一步证明了每个没有$k \cdot K_3$的$q$-粗模型的图都拟等距于一个没有$k \cdot K_3$子式的图，且该拟等距可以仅具有加性失真。这一结果对$k = \infty$也成立。我们还获得了关于两两相距很远的长的导出环的埃尔德什-波萨定理。
英文摘要
We prove the coarse Erdős-Pósa conjecture of Georgakopoulos and Papasoglu. Informally, any graph either contains many fat cycles that are pairwise far apart, or there is a small number of bounded radius balls that together hit all of them. To be more precise, if $G$ is a graph with no $q$-fat model of $k \cdot K_3$ for some $q, k \in \mathbb{N}$, then there is a set $X\subseteq V(G)$ of $\mathcal{O}(k\log k)$ vertices such that every $q$-fat model of $K_3$ in $G$ has distance $\mathcal{O}(q)$ from $X$. In another form more closely resembling Manning's theorem that characterises quasi-trees: if $G$ is a graph with no $q$-fat model of $k \cdot K_3$ for some $q, k \in \mathbb{N}$, then $G$ is $\mathcal{O}(q)$-quasi-isometric to a graph $H$ that contains a set $X\subseteq V(H)$ of $\mathcal{O}(k\log k)$ vertices such that $H-X$ is a forest. Bootstrapping this result, we further prove that every graph with no $q$-fat model of $k \cdot K_3$ is quasi-isometric to a graph with no $k \cdot K_3$ minor, where the quasi-isometry can be chosen to only have additive distortion. This result also holds for $k = \infty$. We also obtain an Erdős-Pósa theorem for long induced cycles that are far apart.
