In 1996, Reed, Robertson, Seymour and Thomas [Combinatorica 1996] proved Younger's Conjecture, which states that, for all directed graphs $D$, there exists a function $f$ such that, if $D$ does not contain $k$ disjoint cycles, then $D$ contains a feedback vertex set, i.e.~a subset of vertices whose deletion renders the graph acyclic, of size bounded by $f(k)$. However, the function obtained by Reed, Robertson, Seymour and Thomas in their paper is enormous and, in fact, not even elementary. We prove the first elementary upper bound for the function $f$ above, showing it is upper-bounded by a power tower of height 8. Our proof is inspired by the breakthrough result of Chekuri and Chuzhoy [J. ACM 2016], who proved a polynomial bound for the Excluded Grid Theorem for undirected graphs. We translate a key concept of their proof to directed graphs by introducing paths of well-linked sets (PWS), and show that any digraph of large directed treewidth contains a large PWS, which in turn contains a large fence. We believe that the theoretical tools developed in this work may find applications beyond the results above, in a similar way as the path-of-sets-system framework due to Chekuri and Chuzhoy [J. ACM 2016] did for undirected graphs (see, for example, Hatzel, Komosa, Pilipczuk and Sorge [Discret. Math. Theor. Comput. Sci. 2022], Chekuri and Chuzhoy [SODA 2015] and Chuzhoy and Nimavat [arXiv 2019]). Indeed, in a follow-up paper, we apply this framework to improve the bounds of the Directed Grid Theorem.

翻译：1996年，Reed、Robertson、Seymour和Thomas [Combinatorica 1996] 证明了Younger猜想，该猜想断言：对于所有有向图$D$，存在函数$f$，使得若$D$不包含$k$个不相交的环，则$D$包含一个反馈顶点集（即删除该顶点子集可使图变为无环图），其大小受$f(k)$限制。然而，Reed、Robertson、Seymour和Thomas在其论文中获得的函数极其巨大，甚至不是初等函数。我们首次证明了上述函数$f$具有初等上界，表明其上界为一个高度为8的幂塔。我们的证明受到Chekuri和Chuzhoy [J. ACM 2016] 突破性成果的启发，他们证明了无向图排除网格定理的多项式界。通过引入良联集路径（PWS）这一关键概念，我们将他们证明中的核心思想移植到有向图中，并证明任何具有大有向树宽的有向图都包含一个大的PWS，而该PWS又包含一个大的栅栏结构。我们相信，本工作发展的理论工具可能超越上述结果得到更广泛的应用，正如Chekuri和Chuzhoy [J. ACM 2016] 提出的集系路径框架在无向图领域所产生的深远影响（参见Hatzel、Komosa、Pilipczuk和Sorge [Discret. Math. Theor. Comput. Sci. 2022]、Chekuri和Chuzhoy [SODA 2015] 以及Chuzhoy和Nimavat [arXiv 2019] 等后续研究）。事实上，在后续论文中，我们将应用此框架改进有向网格定理的界。