Homogeneous structures, bipartite graphs, and partial orders

这个项目主要是在组合学，特别是在研究图和有向图。它们是由一组(点)和边形成的，其中任何两点之间可能有边，也可能没有边。对于有向图，沿边的方向很重要，对于图，则不是这样。如果一个图被细分成两个部分，使得仅有的边在不同部分的点之间，那么它就称为二部图。我们对充分对称的结构感兴趣，也就是说，粗略地说，两个看起来相同的有限子集，意味着它们具有相同的边的大小和排列，但从整个结构的角度来看，它们看起来仍然相同。充分对称的各种结构(如图、有向图、序)的分类在最近的组合学和模型论的工作中得到了突出的体现，这有助于更好地理解这些结构。这项研究将建立在提出者和研究学生关于偏序的早期工作的基础上，而偏序、二部图和有向图之间的联系构成了一个中心主题。从偏序出发的技巧将被应用到上述程序中，以导出2-弧传递二部图的分类和充分对称的有向图。

This project is principally in combinatorics, and most particularly in the study of graphs and digraphs. These are formed from a set (of 'points') and edges, where between any two points there may or may not be an edge. For digraphs, the direction along the edge is important, for graphs, not so. If a graph subdivides into two parts such that the only edges are between points in different parts, then it is said to be bipartite.We are interested here in sufficiently 'symmetric' structures, that is, roughly, those for which two finite subsets which look the same, meaning that they have the same size and arrangements of edges, still look the same from the perspective of the whole structure. Classifications of all the structures of various kinds (such as graphs, digraphs, orders) which are sufficiently symmetric have featured prominently in recent work on combinatorics and model theory, and this has helped to achieve a better understanding of these structures.The research will build on earlier work of the proposer and research students about partial orders, and the links between partial orders, bipartite graphs, and digraphs, form a central theme. Techniques derived from partial orders will be applied to the above programme, to derive classifications for 2-arc transitive bipartite graphs, and sufficiently symmetric digraphs.