Binary Primitive Homogeneous One-based Structures

Suppose that M is countable, binary, primitive, homogeneous, simple and 1-based. We prove that the SU-rank of the complete theory of M is 1. It follows that M is a random structure. The conclusion that M is a random structure does not hold if the binarity condition is removed, as witnessed by the generic tetrahedronfree 3-hypergraph. However, to show that the generic tetrahedron-free 3-hypergraph is 1-based requires some work (it is known that it has the other properties) since this notion is de ned in terms of imaginary elements. This is partly why we also characterize equivalence relations which are de nable without parameters in the context of ω-categorical structures with degenerate algebraic closure. Another reason is that such characterizations may be useful in future research about simple (nonbinary) homogeneous structures.