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Independence, Amalgamation, and Trees


This thesis is concerned with developing a theory of model-theoretic tree properties. These properties are combinatorial properties of a formula or family or formulas that place strong constraints on the behavior of forking and dividing yet are compatible with certain forms of model-theoretic randomness. The most significant and intensively studied is \emph{the tree property}, whose negation characterizes the simple theories, and a successful theory for simple theories was developed by Hrushovski, Kim, Pillay, and others, in the late 90s and early 2000s. Motivated by parallels with simplicity theory, we introduce a theory of independence called \emph{Kim-independence} and present a structure theory for NSOP$_{1}$ theories in terms of it. This unifies and explains simplicity-like phenomena observed in several non-simple examples, such as existentially closed vector spaces with a bilinear form and $\omega$-free PAC fields. This machinery also gives a streamlined method for establishing that a given theory is NSOP$_{1}$ and for showing that certain generic constructions preserve NSOP$_{1}$. We also develop techniques for the manipulation of tree indiscernibles to address several questions concerning the syntax of the related model-theoretic tree properties TP$_{1}$, weak $k$-TP$_{1}$, and the associated cardinal invariants.

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