On the Model Theory of Random Graphs
Brody, Justin D.
Laskowski, Michael C
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Hrushovski's amalgamation construction can be used to join a collection of finite graphs to produce a ``generic'' of this collection. The choice of the collection and the way they are joined are determined by a real-valued parameter α. Classical results have shown that for α irrational in (0,1), the model theory of the resulting structure is very well-behaved. This dissertation examines analogous constructions for rational <italic>r</italic>. Depending on the way in which the parameter's control of the construction is defined, the model theory of the resulting generic will be either very well-behaved or very wild. We characterize when each of these situations occurs.