A set $S \subseteq \omega_3$ is stationary in $\omega_3$ if it intersects all closed and unbounded subsets in $\omega_3$. We say that $S$ reflects below $\omega_3$ if there is some $\alpha<\omega_3$ for which $S\cap \alpha$ is stationary in $\alpha$.

In my thesis I obtain a model $M$ for which every subset of $\omega_3$ focusing on cofinality $\omega$ reflects at a point of cofinality $\omega_1$. That is, $M\models$ Refl$(\omega_3, \omega, \omega_1)$. To do this, I adjust the methods of Gitik in order to get set reflection. This argument may help show the way to getting the consistency strength.

Due to existing bounds, our strategy begins with $\kappa$ of Mitchell order $\kappa$ which concentrates on $\alpha^+$-supercompact cardinals $\alpha$. We collapse $\kappa$ to $\omega_3$, change the cofinality of many measurables between $\omega_2$ and $\omega_3$, and shoot clubs through the non-reflecting stationary sets. For any non-reflecting stationary set $S$, we use the forcing to add a club $C$ for which $C\cap S=\emptyset$, thus rendering this $S$ no longer stationary. We take care that any more stationary sets we add in this process, have a club shot through them and so are no longer stationary in the final generic extension. Additionally, $S\subseteq S^{\omega_3}_\omega$ sets can only have a club shot through them under certain circumstances. In our situation, we use the fact that the complement of these non-reflecting stationary sets will be fat, i.e. for any club $C \subseteq \alpha$, $E\cap C$ contains closed sets of ordinals of arbitrarily large order-type below $\alpha$, to allow us to shoot a club without collapsing $\kappa$.

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.

The main result of this dissertation is a covering theorem for the core model below a Woodin cardinal. More precisely, we work with Steel's core model $\K$ constructed in $V_\Omega$ where $\Omega$ is measurable. The theorem is in a similar spirit to theorems of Mitchell and Cox and roughly says that either $\K$ recognizes the singularity of an ordinal $\kappa$ or else $\kappa$ is measurable in $\K$.

The first chapter of the thesis builds up the technical theory we will work in. The premice we work with use Mitchell-Steel indexing, but we use Jensen's $\Sigma^*$ fine structure and a different amenable coding. The use of $\Sigma^*$ fine structure and this amenable coding significantly simplifies the theory. Towards the end of the first chapter, we prove the full condensation lemma for premice with Mitchell-Steel indexing. This was originally proven by Jensen for premice with $\lambda$-indexing. The second chapter is devoted to the proof of the above mentioned covering theorem.

The purpose of this thesis is to use the tools of Inner Model Theory to the study of notions relative to generic embeddings induced by ideals. We seek to apply the Core Model Induction technique to obtain lower bounds in consistency strength for a specific Stationary Catching principle called StatCatch*(I), related to the saturation of an ideal I of omega2. This principle involves the central notion of self-genericity in its formulation, introduced by Foreman, Magidor and Shelah. In particular, we show that assuming StatCatch*(I) (plus some additional hypothesis in the universe), we can obtain, for every finite number n, an inner model with n Woodin Cardinals.

Ulam proved that there cannot exist a probability measure on the reals for which every set is measurable and gets either measure zero or one. He asked how large a collection of partial 0-1 valued measures is required so that every set of reals is measurable in one of them. Alaoglu and Erdos proved that if the continuum hypothesis holds, then countably many measures is not enough, and Ulam asked if aleph_1 many can suffice. This question was shown to be independent of ZFC by Prikry and Woodin. Here, we examine the analogous questions on successor cardinals above aleph_1 and on spaces of the form P_kappa(lambda). We generalize Woodin's consistency results to these contexts, producing models of ideals of minimal density on various spaces starting from models of almost-huge cardinals. We show some interactions between these ideals, cardinal arithmetic, and square principles. Then we show that certain characterizations of a positive answer to Ulam's question, namely the existence of dense ideals and nonregular ideals, are equivalent on aleph_1 but not for higher cardinals. Some tension appears in separating these properties while preserving the GCH, but we show this is possible using structures we call "coherent forests," about which we show several results of independent interest. The main result is that if almost-huge cardinals are consistent, then ZFC+GCH does not prove that the existence of dense and nonregular ideals is equivalent for successor cardinals above aleph_1. Our methods also lead to a new result on the individual consistency but collective inconsistency of some types of generic large cardinals.

In 1958, Sierpi nski asked whether there exists a linear order $X$ that is isomorphic to its lexicographically ordered cube but is not isomorphic to its square. The corresponding question has been answered positively for many different classes of structures, including groups, rings, graphs, Boolean algebras, and topological spaces of various kinds. However, the main result of this thesis is that the answer to Sierpi nski's question is negative: every linear order $X$ that is isomorphic to its cube is already isomorphic to its square. More generally, if $X$ is isomorphic to any one of its finite powers $X^n$, $n > 1$, it is isomorphic to all of them.

The proof relies on a general representation theorem that characterizes, for a fixed structure $A$ from a class of structures $\mathfrak{C}$, those structures $X \in \mathfrak{C}$ that satisfy the isomorphism $A \times X \cong X$. This characterization is based on an analysis of an arbitrary bijection $f: A\times X \rightarrow X$, and is closely connected to the tail-equivalence relation on the Baire space $A^{\omega}$.

In Chapter 1, we study the tail-equivalence relation as well as those continuous maps on $A^{\omega}$ that preserve tail-equivalence. In Chapter 2, we give our characterization of the isomorphism $A \times X \cong X$, and specify it for several particular classes of structures, including the class of linear orders in which we are primarily interested. In Chapter 3, we use this characterization to solve Sierpi nski's problem, as well as several other problems concerning the multiplication of linear orders. In Chapter 4, we solve a related problem, also due to Sierpi nski, by showing there exist non-isomorphic orders $X$ and $Y$ that divide one another on both sides.

The existence of a saturated ideal on $\aleph_2$ and the $\aleph_2$-tree property are interesting properties in their own right, with the literature filled with results involving one or the other. However, it seems that all known techniques for producing a model with one of these two properties destroys the other property. The question of constructing a model with both properties has appeared in the literature.

In Chapter 1, we cover the technical background needed throughout the rest of the dissertation. In Chapter 2, we present a variant of the Kunen-Magidor construction and use it to construct a model that has a saturated ideal on $\aleph_2$ and satisfies $\square_{\omega_1}$. In Chapter 3, we prove that $\square_{\omega_1, \omega}$ and $\square(\omega_2)$ fail in the original Kunen-Magidor model, and we also show that the original Kunen-Magidor model has simultaneous stationary set reflection. The question of constructing a model that has a presaturated/strong ideal---instead of a saturated ideal---on $\aleph_2$ and has the $\aleph_2$-tree property may be more approachable because it seems that some variant of the Kunen-Magidor construction \textit{may} work in the presaturated/strong case. Thus, in Chapter 4, we present another variant of the Kunen-Magidor construction and use it to prove a partial result towards the goal of constructing a model that has a presaturated/strong ideal on $\aleph_2$ and has the $\aleph_2$-tree property. We then discuss a potential strategy for using the variant to construct such a model.