The theories of Baldwin-Shi hypergraphs and their atomic models

TL;DR

This paper extends the quantifier elimination results of Shelah-Spencer theories to Baldwin-Shi hypergraphs, providing conditions for atomic models and exploring the relationship between existentially closed and atomic models.

Contribution

It introduces Baldwin-Shi hypergraphs as analogues of sparse random graphs and establishes a framework for their model-theoretic properties, including atomic models.

Findings

Quantifier elimination extends to Baldwin-Shi hypergraphs.

A necessary and sufficient condition for atomic models is provided.

Existentially closed models coincide with atomic models in certain classes.

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.