A Coloring Theorem for Inaccessible Cardinals A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Douglas J. Hoffman December 2013 c 2013. Douglas J. Hoffman. All Rights Reserved. 2 This dissertation titled A Coloring Theorem for Inaccessible Cardinals by DOUGLAS J. HOFFMAN has been approved for the Department of Mathematics and the College of Arts and Sciences Todd Eisworth Associate Professor of Mathematics Robert Frank Dean, College of Arts and Sciences 3 Abstract HOFFMAN, DOUGLAS J., Ph.D., December 2013, Mathematics A Coloring Theorem for Inaccessible Cardinals (73 pp.) Director of Dissertation: Todd Eisworth This dissertation proves a coloring theorem for inaccessible cardinals having a stationary subset S ⊂ fδ < κ : cf(δ) = λg that does not reflect in any inaccessible cardinal for a fixed regular uncountable cardinal λ. The theorem shows we can give 2 an \approximate" yes to κ 6! [κ]κ. Recent work of Eisworth has produced partition theorems in the case of successors of singular cardinals, and this dissertation extends his ideas to inaccessible cardinals with a stationary subset that does not reflect in any inaccessible cardinal and consists of singular cardinals of uncountable cofinality. Club- guessing sequences, minimal walks and elementary submodels are used in the proof of the theorem. First, the existence of a nice S-club guessing sequence is established κ when S ⊆ Sλ . After proving the existence of this club-guessing sequence, we look at constructing a club-guessing sequence when S consists of inaccessible cardinals and C is an S-club sequence satisfying ⊗C . Then, we study weakly normal ideals and the effects of a least function on an ideal. The behavior of a least function can have a strong affect on the ideal. When a least function fails to be regressive in a rather strong fashion, this failure forces the ideal to be weakly normal. After investigating weakly normal ideals, the ideal idp(C; I) is introduced, and a coloring is constructed to prove the main theorem in this paper. This coloring has the property that the range of any unbounded subset of κ is almost κ. Using this coloring, we show the 2 ideal idp(C; I) is weakly κ-saturated if κ ! [κ]κ holds. 4 To my wife, Beth, and to my parents, Michael and Lynnae 5 Acknowledgments I would like to thank my wife, Beth, for her love and support through the years. She has been with me for this stage of my life. From my first day of graduate school to the writing of this paper, she has been with me every step of the way and supporting me any way she can. It takes a lot of patience to hear someone talk about something only a few people would find interesting. I would like to thank my parents, Michael and Lynnae Hoffman. They let me do my own thing and taught me to always give it your best effort. Without their support, none of this would be possible. Also, I would like to express my gratitude to my advisor, Dr. Todd Eisworth. His help and advice was not limited to the writing of this dissertation. He got me interested in combinatorial set theory and introduced me to partition theorems. With his help, I was able to explore a rich fragment of combinatorial set theory and con- tribute to the theory. Under his guidance, I was able to go from a graduate student in mathematics to a mathematician. I would also like to thank the members of my dissertation committee for investing time in this manuscript. Chris Baily-Brown, Tammy Matson and Kelly Pero have my gratitude. They have helped me numerous times over the years with forms and deadlines I would have forgotten about. Lastly, I would like to thank Karen Ernst at Hawkeye Community College. She showed me that it was okay to be enthusiastic about mathematics, and doing so, she set me on the path to become a mathematician. 6 Table of Contents Abstract..............................3 Dedication..............................4 Acknowledgments...........................5 1 Introduction...........................7 1.1 What Has Been Done So Far..................7 1.2 Goals For The Paper..................... 15 2 Existence of Nice Club-Guessing Sequences.............. 17 2.1 Club-Guessing Sequences................... 17 2.2 Nice Club-Guessing Sequences: Uncountable Cofinality....... 23 2.3 Constructing A Club-Guessing Sequence Using ⊗C ........ 31 3 Weakly Normal Ideals....................... 37 3.1 An Introduction to Ideals and Filters.............. 37 3.2 Introduction to Weak Normality................ 39 3.3 Weak Normality Arises.................... 41 4 Partition Results......................... 55 4.1 Motivation For The Main Theorem............... 55 4.2 Minimal Walks........................ 55 4.3 The Ideal idp(C; I)...................... 59 4.4 The Main Theorem...................... 63 Bibliography............................. 72 7 1 Introduction 1.1 What Has Been Done So Far The ultimate goal of this dissertation is prove a coloring theorem for certain infinite cardinals. In particular, our focus will be on the negation of the square- bracket relation in the realm of inaccessible cardinals. There has been a lot of work done with the negation of the square-bracket relation. Todorˇcevi´c'swork in [15] describes what happens in the case of the successor of a regular cardinal. Eisworth's work [5, 6, 4] has produced results in the case of successors of singular cardinals. Chapters III and IV of [13] are dedicated to square-bracket partition relations on inaccessible cardinals. Our goals for this paper focus on extending the techniques used on successors of singular cardinals to inaccessible cardinals. Our work builds on a long line of research. In [15], Todorˇcevi´ccreated his method of minimal walks and established a relation between stationary reflection and the failure of a square-bracket relation. Shelah created a plethora of tools utilized in the case of the successor of a singular cardinal. Such tools include club-guessing sequences, club-guessing ideals and scales. In a series of papers, Eisworth has obtained many results dealing with the successors of a singular cardinals. Eisworth's result in [4] is Todorˇcevi´c-like; it relates the failure of a square-bracket relation to stationary reflection. To understand the failure of the square-bracket relation, we start with a brief look at ordinary partition relations. Then, we will see that the failure of an ordinary partition relation leads us to a weak form of homogeneity: the square-bracket relation. From there, we look at the results of Todorˇcevi´cand Eisworth, and end with the goals of this paper. 8 To begin, we look at the following famous theorem in combinatorial set theory: Ramsey's Theorem. THEOREM 1.1.1 (Ramsey's Theorem, [11]) Suppose c :[!]2 ! f0; 1g. Then 2 there is an H ⊆ ! such that f [H] is constant. The function c is referred to as a coloring, and the subset H is said to be homo- geneous for c. A natural question to ask is to what extent does Ramsey's theorem generalize to uncountable cardinals? Before we consider generalizations of Ramsey's theorem, we will remind the reader of arrow notation introduced by Erd}osand Hajnal in [8]. DEFINITION 1.1.1 Let γ be an ordinal, and let κ, λ, τ be cardinals. The symbol λ κ ! (τ)γ is used to indicate that the following assertion holds: For an arbitrary set X of cardinality κ and for every γ-partition f :[X]λ ! γ, there is a set Y ⊆ X and ordinal ν < γ such that jY j = τ and f"[Y ]λ = fνg: Relations of this form are called ordinary partition relations. Ramsey's theo- rem can be restated using this notation. THEOREM 1.1.2 (Ramsey's Theorem) 2 ! ! (!)2 When an ordinary partition relation holds, we know that no matter how [X]λ is partitioned into γ pieces, there will always be a set Y ⊆ X of size τ and ν < γ such that set [Y ]λ is contained in the νth piece of the partition. When dealing with 9 partition relations, we sometimes think of a partition of [κ]λ as a coloring of subsets of κ of size λ. Back to generalizing Ramsey's theorem. One can ask if there are other cardinals for which Ramsey's theorem holds? Equivalently, is there an uncountable cardinal κ for which 2 κ ! (κ)2 @0 2 holds? Sierpi´nksiproved it does not hold in general by showing 2 6! (@1)2. His proof was generalized to show that if Ramsey's theorem holds for an un- countable κ, then κ has to be a strong limit cardinal. Another easy argument shows 2 that Ramsey's theorem fails for singular cardinals, so if κ ! (κ)2, then κ has to be 2 strongly inaccessible. Much more is known: uncountable cardinals for which κ ! (κ)2 are called weakly compact, and they have been studied extensively. Now, we consider the situation when an ordinary partition relation fails. When λ λ κ ! (µ)2 fails, there is a way to partition [κ] into two sets such that for any Y ⊆ κ with jY j = µ,[Y ]λ intersects both parts of the partition. Said another way, there is a function c :[κ]2 ! f0; 1g such that for any Y ⊆ κ with jY j = µ, we have λ rng(f [A] ) = f0; 1g: When an ordinary partition relation fails, it says the partition is complicated in the sense it takes on both colors when restricted to each unbounded subset. If we increase the number of colors, we get a more drastic failure of Ramsey's theorem.