Random structures and automorphisms with a single orbit

We investigate the class of m-hypergraphs whose substructures with l elements have more than s m-element subsets that do not form a hyperedge. The class will have the free amalgamation property if s is small, but it does not if s is large. We find the boundary of s. Suppose the class has the free amalgamation property. In the case \(m \ge 3\) $m\ge 3$ , we demonstrate that the random structure for the class has continuum-many automorphisms with a single orbit. The situation differs from the case of Henson graphs. In the case of generic hypergraphs constructed by Hrushovski’s method using a predimension function, we also demonstrate that they have no automorphisms with a single orbit.