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 "r." 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.