In 2015, Kawarabayashi and Kreutzer proved the Directed Grid Theorem - the generalisation of the well-known Excluded Grid Theorem to directed graphs - confirming a conjecture by Reed, Johnson, Robertson, Seymour and Thomas from the mid-nineties. The theorem states that there is a function $f$ such that every digraph of directed treewidth $f(k)$ contains a cylindrical grid of order $k$ as a butterfly minor. However, the given function grows faster than any non-elementary function of the size of the grid minor. More precisely, it is larger than a power tower whose height depends on the size of the grid. In this paper, we present an alternative proof of the Directed Grid Theorem which is conceptually much simpler, more modular in composition and improves the upper bound for the function $f$ to a power tower of height $22$. A key concept of our proof is a new structure called cycles of well-linked sets (CWS). We show that any digraph of large directed treewidth contains a large CWS, which in turn contains a large cylindrical grid.

翻译：2015年，Kawarabayashi 与 Kreutzer 证明了有向网格定理——这是著名的禁止网格定理在有向图上的推广，从而证实了 Reed、Johnson、Robertson、Seymour 和 Thomas 在九十年代中期提出的猜想。该定理指出，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 $k$ 的柱面网格作为蝴蝶子式。然而，原证明给出的函数增长速度超过了网格子式大小的任何非初等函数。更准确地说，它大于一个高度依赖于网格大小的幂塔。在本文中，我们提出了有向网格定理的一个替代证明，该证明在概念上更为简单，结构上更具模块化，并将函数 $f$ 的上界改进为高度为 $22$ 的幂塔。我们证明的一个关键概念是一种称为良连通集环的新结构。我们证明，任何具有大有向树宽的有向图都包含一个大的良连通集环，而该环又包含一个大的柱面网格。