In 2015, Kawarabayashi and Kreutzer proved the directed grid theorem confirming a conjecture by Reed, Johnson, Robertson, Seymour, and Thomas from the mid-nineties. The theorem states the existence of a function $f$ such that every digraph of directed tree-width $f(k)$ contains a cylindrical grid of order $k$ as a butterfly minor, but the given function grows non-elementarily with the size of the grid minor. In this paper we present an alternative proof of the directed grid theorem which is conceptually much simpler, more modular in its composition and also improves the upper bound for the function $f$ to a power tower of height 22. Our proof is inspired by the breakthrough result of Chekuri and Chuzhoy, 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 \emph{cycles of well-linked sets (CWS)}, and show that any digraph of high directed tree-width contains a large CWS, which in turn contains a large cylindrical grid, improving the result due to Kawarabayashi and Kreutzer from an non-elementary to an elementary function. An immediate application of our result is an improvement of the bound for Younger's conjecture proved by Reed, Robertson, Seymour and Thomas (1996) from a non-elementary to an elementary function. The same improvement applies to other types of Erd\H{o}s-P\'osa style problems on directed graphs. To the best of our knowledge this is the first significant improvement on the bound for Younger's conjecture since it was proved in 1996. We believe that the theoretical tools we developed may find applications beyond the directed grid theorem, in a similar way as the path-of-sets-system framework due to Chekuri and Chuzhoy (2016) did (see for example Hatzel, Komosa, Pilipczuk and Sorge (2022); Chekuri and Chuzhoy (2015); Chuzhoy and Nimavat (2019)).

翻译：2015年，Kawarabayashi和Kreutzer证明了有向网格定理，证实了Reed、Johnson、Robertson、Seymour和Thomas在90年代中期提出的一个猜想。该定理断言存在一个函数$f$，使得每个有向树宽为$f(k)$$的有向图都包含一个阶为$k$的圆柱网格作为蝴蝶子式，但该函数随网格子式尺寸的增长呈非初等增长。本文提出有向网格定理的另一种证明，该证明在概念上更为简洁、组成上更具模块性，并将函数$f$的上界改进为高度为22的幂塔。我们的证明受Chekuri和Chuzhoy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。我们通过引入\emph{良连通集循环（CWS）}将他们证明的关键概念迁移至有向图，并证明任何具有高有向树宽的有向图都包含一个大型CWS，而该CWS又包含一个大型圆柱网格，从而将Kawarabayashi和Kreutzer的结果从非初等函数改进为初等函数。我们的结果的一个直接应用是将Reed、Robertson、Seymour和Thomas（1996年）证明的Younger猜想的上界从非初等函数改进为初等函数。同样的改进也适用于有向图上的其他类型Erdős-Pósa型问题。据我们所知，这是自1996年Younger猜想被证明以来，其界取得的首次重大改进。我们相信，所发展的理论工具可能找到超出有向网格定理的应用，类似于Chekuri和Chuzhoy（2016年）提出的路径集系统框架所发挥的作用（例如参见Hatzel、Komosa、Pilipczuk和Sorge（2022年）；Chekuri和Chuzhoy（2015年）；Chuzhoy和Nimavat（2019年））。