Let ${\cal G}$ and ${\cal H}$ be minor-closed graph classes. The pair $({\cal H},{\cal G})$ is an Erd\H{o}s-P\'osa pair (EP-pair) if there is a function $f$ where, for every $k$ and every $G\in{\cal G},$ either $G$ has $k$ pairwise vertex-disjoint subgraphs not belonging to ${\cal H},$ or there is a set $S\subseteq V(G)$ where $|S|\leq f(k)$ and $G-S\in{\cal H}.$ The classic result of Erd\H{o}s and P\'osa says that if $\mathcal{F}$ is the class of forests, then $({\cal F},{\cal G})$ is an EP-pair for every ${\cal G}$. The class ${\cal G}$ is an EP-counterexample for ${\cal H}$ if ${\cal G}$ is minimal with the property that $({\cal H},{\cal G})$ is not an EP-pair. We prove that for every ${\cal H}$ the set $\mathfrak{C}_{\cal H}$ of all EP-counterexamples for ${\cal H}$ is finite. In particular, we provide a complete characterization of $\mathfrak{C}_{\cal H}$ for every ${\cal H}$ and give a constructive upper bound on its size. Each class ${\cal G}\in \mathfrak{C}_{\cal H}$ can be described as all minors of a sequence of grid-like graphs $\langle \mathscr{W}_{k} \rangle_{k\in \mathbb{N}}.$ Moreover, each $\mathscr{W}_{k}$ admits a half-integral packing: $k$ copies of some $H\not\in{\cal H}$ where no vertex is used more than twice. This gives a complete delineation of the half-integrality threshold of the Erd\H{o}s-P\'osa property for minors and yields a constructive proof of Thomas' conjecture on the half-integral Erd\H{o}s-P\'osa property for minors (recently confirmed, non-constructively, by Liu). Let $h$ be the maximum size of a graph in ${\cal H}.$ For every class ${\cal H},$ we construct an algorithm that, given a graph $G$ and a $k,$ either outputs a half-integral packing of $k$ copies of some $H \not\in {\cal H}$ or outputs a set of at most ${2^{k^{\cal O}_h(1)}}$ vertices whose deletion creates a graph in ${\cal H}$ in time $2^{2^{k^{{\cal O}_h(1)}}}\cdot |G|^4\log |G|.$

翻译：令${\cal G}$和${\cal H}$为对子式封闭的图类。若存在函数$f$使得对于每个$k$和每个$G\in{\cal G}$，要么$G$包含$k$个互不相交（顶点不相交）且不属于${\cal H}$的子图，要么存在顶点子集$S\subseteq V(G)$满足$|S|\leq f(k)$且$G-S\in{\cal H}$，则称$({\cal H},{\cal G})$为Erdős-Pósa对（EP-对）。Erdős和Pósa的经典结果表明：若$\mathcal{F}$为森林类，则对于任意${\cal G}$，$({\cal F},{\cal G})$构成EP-对。若${\cal G}$是满足$({\cal H},{\cal G})$不构成EP-对这一性质的最小图类，则称${\cal G}$为${\cal H}$的EP-反例。我们证明对于任意${\cal H}$，其所有EP-反例构成的集合$\mathfrak{C}_{\cal H}$是有限的。具体而言，我们完整刻画了任意${\cal H}$对应的$\mathfrak{C}_{\cal H}$，并给出了其大小的构造性上界。每个${\cal G}\in \mathfrak{C}_{\cal H}$均可描述为一系列网格状图$\langle \mathscr{W}_{k} \rangle_{k\in \mathbb{N}}$的所有子式。此外，每个$\mathscr{W}_{k}$允许半整填充：存在某个$H\not\in{\cal H}$的$k$个副本，其中每个顶点至多被使用两次。这完整划定了Erdős-Pósa性质对于子式的半整性阈值，并为Thomas关于子式半整Erdős-Pósa性质的猜想（近期由Liu非构造性地证实）提供了构造性证明。令$h$为${\cal H}$中图的最大规模。对于任意图类${\cal H}$，我们构造了算法：给定图$G$和参数$k$，该算法要么输出某个$H \not\in {\cal H}$的$k$个副本的半整填充，要么在$2^{2^{k^{{\cal O}_h(1)}}}\cdot |G|^4\log |G|$时间内输出最多${2^{k^{\cal O}_h(1)}}$个顶点的集合，删除这些顶点后得到的图属于${\cal H}$。