Combinatorial structures with lots of symmetry

There are various levels of symmetry that a relational structure, such as a graph, digraph or poset, can display. One of the strongest possible symmetry properties is homogeneity: a structure M is called homogeneous if any isomorphism between finite induced substructures of M extends to an automorphism of M. Homogeneity is a concept that arises in model theory, and originally goes back to fundamental work of Fraisse (1953). There are many interesting examples of homogeneous structure, including the amazing "random graph" constructed by Richard Rado (1964). Using the random graph as an illustrative example, in this talk I will give some background on the theory of homogeneous structures, and in particular will survey some of the known classification results. Then I will go on to discuss various natural ways that homogeneity may be weakened, but in such a way that it is still possible for classification results to be obtained.