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.

我们证明了Shelah-Spencer的量化消除结果几乎确定了稀疏随机图\（G（n，n ^ {-\ alpha}）\）的理论Laskowski（Isr J Math 161：157–186，2007）给出的结果扩展到了它们的各种类似物。将获得类似物作为某些类的有限结构的泛型结构的理论，并具有由秩函数引起的强子结构的概念，我们将其称为泛型Baldwin–Shi超图。在此过程中，我们提供了一种构造扩展名的方法，该扩展名的“相对等级”为负，但在上下文中任意较小。我们为Baldwin-Shi超图理论具有原子模型提供了充要条件。我们进一步证明，对于鲍德温-史超图的某些行为良好的理论类别，存在闭式模型和原子模型是相对应的。