Danul K. Gunatilleka | Searcharxiv

arXiv2018

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 $G(n,n^{-α})$ 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.

math.LO↗

arXiv2018

Countable models of the theories of Baldwin-Shi hypergraphs and their regular types

We continue the study of the theories of Baldwin-Shi hypergraphs from $[5]$. Restricting our attention to when the rank $δ$ is rational valued, we show that each countable model of the theory of a given Baldwin-Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class of finite structures with the inherited notion of strong substructure. We introduce a notion of dimension for a model and show that there is a an elementary chain $\{\mathfrak{M}_β:β<ω+1\}$ of countable models of the theory of a fixed Baldwin-Shi hypergraph with $\mathfrak{M}_β\preccurlyeq\mathfrak{M}_γ$ if and only if the dimension of $\mathfrak{M}_β$ is at most the dimension of $\mathfrak{M}_γ$ and that each countable model is isomorphic to some $\mathfrak{M}_β$. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on the work of Brody and Laskowski, we use these structures to give an example of a pseudofinite, $ω$-stable theory with a non-locally modular regular type, answering a question of Pillay in $[9]$.

math.LO↗