Minors at large

Robertson-Seymour图的次要定理是图论中最深入、最重要的结果之一。它断言在次关系下闭合的图族F可以由一个有限的“禁止”子结构列表确定，即一个图G属于F当且仅当G不包含这些子结构中的一个。这一不朽的定理在1983年至2004年发表的一系列长达500多页的20篇论文中得到了证明。证据有许多副作用，对该地区产生了独立的影响。图的次要定理的一个重要含义是，如上所述的每个族F都可以被一个有效的算法识别。此外，许多已知的算法问题通常是NP难的，当限制在这样的族F中时，已知有有效的算法。这个项目是Fujiwara&Papasoglou和Bonamy等人最近工作的后续。发展了一个类似经典图子理论的“粗图次要理论”，但引入了遵循格罗莫夫的粗略几何范例的粗略视角。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。我们设想了一个经典图着色问题的粗略版本，它将在计算几何问题中立即有算法应用。由于几何技术在数据科学中的应用，预计这类问题的重要性将会增加。我们提出了图论中其他经典结果的几何类比。在第二部分中，我们攻击Thomas的一个著名猜想，该猜想试图将Robertson-Seymour图的小定理推广到可数无限的图。我们发现Thomas猜想对有限图也有重要的意义，并提出了一个更弱的形式，它仍然具有这个意义，而且更有可能是可达的。我们提出了打击后者的具体步骤。

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.