Given a sequence {a n } in (0,1) converging to a rational, we examine the model theoretic properties of structures obtained as limits of Shelah-Spencer graphs G(m, m -αn ). We show that in most cases the model theory is either extremely well-behaved or extremely wild, and characterize when each occurs.

We show that the quantifier elimination result for the Shelah-Spencer almost sure theories of sparse random graphs $$G(n,n^{-\alpha })$$ given by Laskowski (Isr J Math 161:157–186, 2007) extends to their various analogues. The analogues will be obtained as theories of generic structures of certain classes of finite structures with a notion of strong substructure induced by rank functions and we will call the generics Baldwin–Shi hypergraphs. In the process we give a method of constructing extensions whose ‘relative rank’ is negative (...) but arbitrarily small in context. We give a necessary and sufficient condition for the theory of a Baldwin–Shi hypergraph to have atomic models. We further show that for certain well behaved classes of theories of Baldwin–Shi hypergraphs, the existentially closed models and the atomic models correspond. (shrink)