Extremal Problems Linking Graphs and Set Systems

极值集理论中的许多问题和结果，可以通过在基集上施加某种图结构来推广。这往往会导致有趣的问题，涉及到图论和其他组合技术的混合。我们将首先考虑以这种方式扩展分离集系统的概念。现有的工作在这个方向上主要涉及分离路径，即使在这里也有一些有趣的开放问题。可以提出的问题涉及特殊的图族、一般的边界或比路更复杂的结构，这些方向中的每一个都有不同的味道。图和集合系统之间的另一个接口是离散超立方体。在这里，我们建议工作的问题有关的路径和循环结构，一个出发点是诺林的对映着色猜想。我们还提出了一些想法的主管和尼古拉斯日，Voronoi游戏在超立方体，这带来了问题，从度量几何到超立方体设置。

Many problems and results from extremal set theory, can be generalised by imposing some graph structure on the ground set. This often leads to interesting questions which involve a mixture of graph theoretic and other combinatorial techniques. We will initially consider extending the concept of separating set systems in this way. Existing work in this direction mainly involves separating paths and even here there are some intriguingly open problems. Questions can be asked involving particular graph families, general bounds, or more complicated structures than paths, and each of these directions has a different flavour.Another point of interface between graphs and set systems is the discrete hypercube. Here, we propose to work on problems concerning the path and cycle structure, a starting point being Norine's antipodal colouring conjecture. We also propose to develop some ideas of the supervisor and Nicholas Day, on Voronoi games in the hypercube, which brings problems from metric geometry into the hypercube setting.