Minors at large

Robertson-Seymour图子式定理是图论中最深刻和最重要的结果之一。它断言，在次关系下封闭的图族F可以由有限的“禁止”子结构列表确定，在这个意义上，图G属于F当且仅当G不包含这些子结构之一。这个不朽的定理在1983年至2004年发表的20篇论文中得到了证明，这些论文长达500多页。该证明有许多副作用，对该领域产生了独立的影响。图小定理的一个重要含义是上面的每个族F都可以通过有效的算法识别。此外，许多算法问题，被称为是NP-困难的一般被称为有效率的算法时，限制到这样一个家庭F.这个项目跟进藤原和Papasoglou和Bonamy等人最近的工作。重要的是，这个新定理不仅适用于图，而且适用于更广泛的度量空间，包括黎曼流形和计算机科学中出现的许多离散度量空间。我们设想一个粗糙的版本的经典图着色问题，将立即在计算几何问题的算法应用。这些问题的重要性，预计将增长由于就业的几何技术在数据科学。我们提出了图论中其他经典结果的几何类似物。在第二部分中，我们攻击了托马斯的一个众所周知的猜想，该猜想试图将Robertson-Seymour图小定理扩展到可数无限图。我们观察到托马斯猜想对有限图也有重要的意义，并提出了一个更弱的版本，它仍然有这个意义，而且更容易被理解。我们提出了解决后者的具体步骤。

The Robertson-Seymour Graph Minor theorem is one of the deepest and most important results in Graph Theory. It asserts that every family F of graphs that is closed under the minor relation can be determined by a finite list of "forbidden" substructures, in the sense that a graph G belongs to F if and only if G does not contain one of these substructures. This monumental theorem was proved in a series of twenty papers spanning over 500 pages, published from 1983 to 2004. The proof had many side-results that have had an independent impact on the area. An important implication of the Graph Minor theorem is that every family F as above can be recognised by an efficient algorithm. Moreover, many algorithmic problems that are known to be NP-hard in general are known to have efficient algorithms when restricted to such a family F.This project follows up on recent work by Fujiwara & Papasoglou and Bonamy et al. to develop a "Coarse Graph Minor Theory" that parallels the classical graph minor theory but introduces a coarse perspective following Gromov's paradigm of coarse geometry. Importantly, this new theorem applies not only to graphs, but to a much broader classes of metric spaces including Riemannian manifolds and many discrete metric spaces arising in computer science. We envisage a coarse version of a classical graph colouring problem that would have immediate algorithmic applications in problems of computational geometry. The importance of such problems is expected to grow due to the employment of geometric techniques in data science. We propose geometric analogues of other classical results of graph theory. In a second part, we attack a well known-conjecture of Thomas that seeks to extend the Robertson-Seymour Graph Minor Theorem to countably infinite graphs. We objerve that Thomas' conjecture would have an important implication for finite graphs as well, and propose a weaker version that would still have this implication and is much more likely to be accessible. We propose concrete steps for attacking the latter.