The theories of Baldwin-Shi hypergraphs and their atomic models

DOI10.1007/S00153-021-00765-8arXiv1803.01831MaRDI QIDQ6298628

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Publication date: 5 March 2018

Abstract: 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 in $[7]$ 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.

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