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Trees, Refining, and Combinatorial Characteristics

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The analysis of trees and the study of cardinal characteristics are both of historical and contemporary importance to set theory. In this thesis we consider each of these topics as well as questions relating to (almost) disjoint refinements. We show how structural information about trees and other similar objects is revealed by investigating the determinacy of certain two player games played on them. The games we investigate have classical analogues and can be used to prove structural dichotomies and related results. We also use them to find generalizations of the topological notions of perfectness and scatteredness for spaces like $2^{\kappa}$ and $P_{\kappa}\lambda$ and form connections to when a submodel is e.g. ``$T$-guessing" for a certain tree $T$. Questions surrounding generalizations of the cardinal characteristics $\mathfrak{t}$ (the tower number), $\mathfrak{h}$ (the distributivity number), and $\textbf{non}(\mathcal{M})$ (the uniformity number for category) in particular are considered. For example, we ask whether or not $\mathfrak{h}(\kappa)$ can be defined in a reasonable way. We give several impediments. Generalizations of a combinatorial characterization of $\textbf{non}(\mathcal{M})$ in terms of countably matching families of functions become central for our investigation, and we show how characteristics relating to these generalizations can be manipulated by forcing. Similarly, the question of in which contexts can outer models can add strongly disjoint functions is considered. While Larson has shown \cite{Larson2007} that this is possible with a proper forcing at $\omega_1$, and it is a corollary of a result of Abraham and Shelah \cite{AbrahamShelah1986} that it is consistently impossible at $\omega_2$, we note with Radin forcing that if $\kappa$ has a sufficient amount of measurable reflection, then it can be done at $\kappa$. Turning to the theory of disjoint refinements, we generalize a recent result of Brendle \cite{Soukup2008}, and independently Balcar and Paz {a}k \cite{BalcarPazak2010}, that any time a real is added in an extension, the set of ground model reals can be almost disjointly refined to the setting of adding subsets of $\kappa$, and consider related topics.

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