The grid theorem, originally proved by Robertson and Seymour in Graph Minors V in 1986, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance in bidimensionality theory, and it is the basis for several other structure theorems developed in the graph minors project. In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas (see [Reed 97, Johnson, Robertson, Seymour, Thomas 01]), independently, conjectured an analogous theorem for directed graphs, i.e. the existence of a function f : N -> N such that every digraph of directed tree-width at least f(k) contains a directed grid of order k. In an unpublished manuscript from 2001, Johnson, Robertson, Seymour and Thomas give a proof of this conjecture for planar digraphs. But for over a decade, this was the most general case proved for the Reed, Johnson, Robertson, Seymour and Thomas conjecture. Only very recently, this result has been extended to all classes of digraphs excluding a fixed undirected graph as a minor (see [Kawarabayashi, Kreutzer 14]). In this paper, nearly two decades after the conjecture was made, we are finally able to confirm the Reed, Johnson, Robertson, Seymour and Thomas conjecture in full generality and to prove the directed grid theorem. As consequence of our results we are able to improve results in Reed et al. in 1996 [Reed, Robertson, Seymour, Thomas 96] (see also [Open Problem Garden]) on disjoint cycles of length at least l and in [Kawarabayashi, Kobayashi, Kreutzer 14] on quarter-integral disjoint paths. We expect many more algorithmic results to follow from the grid theorem.

翻译：网格理论最初在1986年由Robson和Seymour Prophic Medients V 中由Robertson和Seymour所证明,是图中未成年人研究的最核心结果之一。它在算法图结构理论中发现了许多应用,例如二维理论,它是图中未成年人项目中开发的若干其他结构理论的基础。在90年代中期,Reed和Johnson、Robertson、Seymour和Thomas为图解提供了证据(见[Reed 97、Johnson、Robertson、Seymour、Thomas 01]),独立地,为方向图中图中图中图中图中显示了一个相似的理论,也就是一个相似的理论。在1996年中图中,Remary Robertson、Seyson、Seymmour 01 和Thomas 模型中显示了一个最不长的理论。在1996年中,Remaryal 和Remartal 的成绩最终在Smasal 14 中显示,这个结果在1996年的Smartal 。最后结果在Smaxeal 之后,这个结果在Smart 。在Smartal 中显示,最后在Smartal 14 之后,这个结果在Smartal 。在Smartal ladeal lade 中显示,这个结果在Slade 之后,这个结果在Sta lax 。