Sporadic homogeneous structures

We discuss Lachlan’s classification theory for finite homogeneous structures and related problems on finite permutation groups. Lachlan’s theory provides a hierarchy of classifications in which structures which are “sporadic” in one context reappear as members of infinite families at later stages. Every finite structure is accounted for at some level in this hierarchy, but for structures associated with familiar primitive permutation groups the combinatorial problem of locating that level precisely can be quite challenging. . . . in most categories few objects have the Witt property; those that do are very well behaved in-