Extremal Problems Linking Graphs and Set Systems

极值集合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。这经常导致有趣的问题，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合系统的概念。这方面的现有工作主要涉及分离路径，即使在这里也存在一些有趣的开放问题。问题可以涉及特定的图族、一般边界或比路径更复杂的结构，并且每个方向都有不同的风格。图和集合系统之间的另一个交界点是离散超立方体。在这里，我们建议研究关于路和圈结构的问题，一个起点是Norine的对极着色猜想。我们还提出了Supervisor和Nicholas Day关于超立方体中的Voronoi对策的一些想法，这将度量几何中的问题引入到超立方体环境中。

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.