Homogeneous structures, bipartite graphs, and partial orders

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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.