Random structures and automorphisms with a single orbit

Abstract 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 $$m \ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , 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.