The theories of Baldwin–Shi hypergraphs and their atomic models

We show that the quantifier elimination result for the Shelah-Spencer almost sure theories of sparse random graphs G(n,n-α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(n,n^{-\alpha })$$\end{document} 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.