On the Infinitary Combinatorics of Small Cardinals and the Cardinality of the Continuum

This work is divided into two parts which are concerned, respectively, with the combinatorics of the cardinals $\aleph_1$ and $\aleph_2$. The first part of the thesis contains the result due to the author and his advisor, Itay Neeman, that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum; this answers a long-standing open question in forcing. Most of Part 1 appears in our submitted manuscript \cite{GN}. After surveying the relevant background in the first chapter, we proceed in the second chapter to define the notion of a Partition Product. This is a type of iteration built out of smaller ones in specific ways, roughly with memory conditions on the names and with isomorphism and coherence conditions on the various ``memories." We will prove a number of useful facts about partition products in Chapter 2. In Chapter 3, we show how to construct so-called Preassignments of Colors in the context of partition products; this forms the technical heart of Part 1. And finally, in Chapter 4, we show how to construct partition products in $L$; in particular, we construct the partition product which yields the model witnessing our theorem.Part 2 of the thesis addresses questions about a variety of combinatorial principles on $\aleph_2$. In each chapter in Part 2, we will be concerned with showing that some amount of Stationary Reflection holds at $\omega_2$, more specifically showing that various amounts of stationary reflection are compatible with other principles of wide interest. In Chapter 5, we provide an overview of these combinatorial principles and some of their history; we also spend some time collecting standard facts about Mitchell-type forcings which we will use in the subsequent chapters. Chapter 5 concludes with a proof that various Mitchell-type posets and their quotients are proper, a result which we assume is known, but which we have not encountered ourselves elsewhere.Chapters 6 and 7 address questions arising from the recent paper \emph{The Eightfold Way} by Cummings, Friedman, Magidor, Rinot, and Sinapova (see \cite{8fold}). In Chapter 6, we answer an open question asked at the end of that paper by showing that it is consistent, from a Mahlo cardinal, that the Tree Property $(\TP)$ and Approachability $(\AP)$ both fail at $\om_2$, while stationary reflection ($\RP$) holds at $\omega_2$. The authors of \cite{8fold} obtained the consistency of this same configuration from a weakly compact cardinal; our result proves the consistency of this configuration from optimal assumptions. We remark here that we present the original proof discovered by the author of this thesis. Later, the author, working with John Krueger, provided a more streamlined proof of this same result; this proof will appear in the forthcoming paper \cite{GiltonKrueger8}. Chapter 6 also includes an unrelated Easton-style lemma for preserving stationary subsets of countable cofinality; this result is due to the author and Omer Ben-Neria.In Chapter 7, we show that for any Boolean combination, $\Phi$, of $\TP$ and $\AP$, $\Phi$ is consistent with a strong form of simultaneous stationary reflection on $\omega_2$, namely that every stationary $S\seq\om_2\cap\cof(\om)$ reflects almost everywhere. This strengthens some of the results from \cite{8fold}. In Chapter 8, we return to the model from \cite{GiltonKruegerHS}, making good on a promise from the postscript therein. In \cite{GiltonKruegerHS}, the author and John Krueger originally sought to show that stationary reflection on $\om_2$ is consistent with a large continuum, and we built an involved mixed-support iteration to achieve such a model. However, we later learned from I. Neeman that such a model can be constructed by simply adding Cohen reals over the original Harrington-Shelah model (\cite{HS}). In Chapter 8 we will show that after a modification of our original preparatory iteration, we may obtain a model in which $\RP$ and $\AP$ both hold, in which $2^\om>\om_2$, and in which there are neither special Aronszajn trees on $\om_2$ nor weak Kurepa trees on $\omega_1$. This is a configuration which cannot be obtained simply by adding Cohen reals over the original Harrington-Shelah model nor by the methods of disjoint stationary sequences from \cite{GiltonKrueger8}. We hope that this demonstrates the usefulness of such a mixed support iteration.In the final chapter, we provide a list of open questions which we would like to address in future work