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 ⊂ {δ < κ : cf(δ) = λ} 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 :[ω]2 → {0, 1}. Then
2 there is an H ⊆ ω 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 |Y | = τ and
f”[Y ]λ = {ν}.
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 ω → (ω)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
ℵ0 2 holds? Sierpi´nksiproved it does not hold in general by showing 2 6→ (ℵ1)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 |Y | = µ,[Y ]λ intersects both parts of the partition. Said another way, there is a function c :[κ]2 → {0, 1} such that for any Y ⊆ κ with |Y | = µ, we have
λ rng(f [A] ) = {0, 1}.
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. To study the failure of an ordinary partition relation, we turn to another partition relation called the square-bracket relation. Again, the notation is due to Erd˝os and Hajnal [8].
DEFINITION 1.1.2 Given cardinals κ, λ and µ and ordinal γ, the symbol
λ κ → [µ]γ 10
indicates the following assertion holds: For an arbitrary set X of cardinality κ and for any arbitrary partition f :[X]λ → γ, there is a set Y with |Y | = µ for which f”[Y ]λ 6= γ.
Compared to ordinary partition relations, the square-bracket relation is a weak form of homogeneity. When studying square-bracket relations, we are more interested in when it fails. If we have
λ κ 6→ [µ]γ , then there is a set X of cardinality κ and a partition of f :[X]λ → γ such that for any set Y ⊆ X with |Y | = µ, we have
f”[Y ]λ = γ.
These colorings are so complicated that they take on all possible values whenever the coloring is restricted to a set of size µ. The negation of a square-bracket relation at a cardinal κ yields a strong failure of Ramsey’s theorem by giving us a coloring that remains complicated when restricted to any unbounded subset of κ.
ℵ0 2 Sierpi´nksi’sproof of 2 6→ (ℵ0)2 actually shows
ℵ0 2 2 6→ [ℵ1]2.
In his proof, Sierpi´nksiused an arbitrary well-order of R and the regular ordering
of R to define the coloring. The coloring was defined as follows: for x0 < x1 ∈ R,
f({x0, x1}) = 0 if the well-order and regular order on R agreed, and 1 otherwise. If
2 2 H ⊂ R with |H| = ℵ1 and f [H] = {0} or f [H] = {1}, then H with the regular ordering or reverse ordering would be well-ordered. It can be shown that any such set is at most countable, and we have a contradiction.
ℵ0 2 Sierpi´nksi’sproof of 2 6→ [ℵ1]2 can be generalized as follows.
THEOREM 1.1.3 (Sierpi´nksi,[14]) For each infinite κ ≥ ω,
κ + 2 2 6→ [κ ]2. 11
A later result was obtained by Erd˝osin [8].
THEOREM 1.1.4 If κ ≥ ω and 2κ = κ+, then
+ + 2 κ 6→ [κ ]κ+ .
This theorem differs from the previous two by increasing the number of colors in
+ ℵ0 the partition from 2 to κ . Under the Continuum Hypothesis ( 2 = ℵ1), the above theorem states that
2 ℵ1 6→ [ℵ1]ℵ1 .
The question whether ℵ 6→ [ℵ ]2 holds without the use of the Continuum Hypothesis 1 1 ℵ1 was a major open question in set theory until it was settled by Stevo Todorˇcevi´cin [15].
THEOREM 1.1.5 (Todorˇcevi´c,[15]) Suppose ω < λ = cf(λ) has a non-reflecting stationary subset, then
2 λ 6→ [λ]λ.
A stationary subset S ⊆ κ is non-reflecting if S ∩ α is not stationary in α for all limit α < κ. It can be shown that if κ is a regular cardinal, then the set
κ+ + Sκ := {δ < κ : cf(δ) = κ}
is a non-reflecting stationary subset of κ+. Along with this fact, Todorˇcevi´c’stheorem shows for every regular cardinal κ ≥ ω that
+ + 2 κ 6→ [κ ]κ+ .
In particular, by fixing κ = ℵ0, Todorˇcevi´c’stheorem yields the following corollary.
COROLLARY 1.1.1 (Todorˇcevi´c,[15])
2 ℵ1 6→ [ℵ1]ℵ1 . 12
In Todorˇcevi´c’sproof, he used a non-reflecting stationary subset, S ⊆ λ, to create a sequence C = hCδ : δ < λi such that
i. Cδ is a club subset of δ with 0 ∈ Cδ for each δ < λ, and
ii. for each limit δ < λ, Cδ ∩ S = ∅.
Todorˇcevi´ccalls a sequence of this form a C-sequence. Todorˇcevi´cuses C- sequences to produce, for each pair of ordinals α < β < λ, a decreasing (hence finite) sequence of ordinals beginning with β and ending with α. These sequences are produced using his technique of minimal walks, and he uses them to define a complicated partition of λ which then yields the negative square-bracket partition relation.
κ+ Todorˇcevi´c’stheorem applies to many cardinals but not all of them. Since Sκ is a non-reflecting stationary subset for a regular κ, the question whether a square-bracket relation holds for the successor of a regular cardinal is settled. But what about the successor of a singular cardinal? When working with the successor of a singular cardinal, we run into trouble be- cause it is consistent (assuming large cardinals) that every stationary subset of such cardinal reflects ([4]). In the context of successors of singular cardinals, new tech- niques and tools needed to be developed. Collaborating with Shelah, Eisworth pro- duced some results in this situation. In [5], Eisworth showed when µ is a singular cardinal,
+ + 2 µ 6→ [µ ]µ+ if and only if
+ + 2 µ 6→ [µ ]θ for arbitrary large θ < µ.
And, in [4], Eisworth was able to give concrete conditions for the negation of the square-bracket relation. 13
+ + 2 THEOREM 1.1.6 (Eisworth, [4]) If µ is a singular cardinal and µ → [µ ]µ+ , then every collection of fewer than cf(µ) stationary subsets of µ+ reflects simultane- ously.
In order to get these results in the case of the successor of a singular cardinal, Eisworth used a multitude of mathematical tools. From Shelah, Eisworth used scales and club-guessing sequences, and from Todorˇcevi´c,he used Todorˇcevi´c’smethod of minimal walks. Using these tools, Eisworth constructed a coloring with the property that the range of any unbounded subset of µ+ is a large set relative to a specific ideal. Then, he shows the existence of this ideal is impossible if a square-bracket relation holds. An important tool used in Eisworth’s proof of Theorem 1.1.6 was the S-club guessing sequence. An S-club guessing sequence is a sequence of subsets that can “guess” when a club is club “a lot.” In the definition of an S-club guessing sequence, the terms “guesses” and “a lot” are defined in a rigorous manner.
DEFINITION 1.1.3 Suppose κ is an uncountable cardinal and S ⊆ κ is a station- ary subset. We say hCδ : δ ∈ Si is an S-club guessing sequence if
i. For each δ ∈ S, Cδ ⊆ δ is club in δ.
ii. For each club E ⊆ κ, there are stationarily many δ ∈ S such that Cδ “guesses” E ∩ δ.
There are a few ways that an S-club guessing sequence can “guess” a club. When we want to “guess” a club, we typically require one of the following:
1. Cδ ⊆ E,
2. Cδ \ E is bounded in δ,
3. δ = sup(E ∩ nacc(Cδ)), or 14
4. δ = sup(Cδ ∩ E).
An ordinal α is a non-accumulation point for a set A if α ∈ A and sup(α∩A) < α. For a set A, we denote the set of non-accumulation points of A by nacc(A). Depending on the situation, we may require more from our S-club sequence. For
κ instance, if S ⊆ Sλ for some regular λ < κ, we may require that otp(Cδ) = λ. By
requiring this condition, we can ensure that |Cδ| = λ for each δ ∈ S. To see why S-club guessing sequences are of some interest, we consider the well- known principle ♦(S).
DEFINITION 1.1.4 Let κ be a cardinal and S ⊆ κ be a stationary subset of κ.
♦(S) asserts the existence of a sequence hCδ : δ ∈ Si with Cδ ⊂ δ such that for every X ⊆ κ, there are stationarily many δ ∈ S such that
X ∩ δ = Cδ.
Notice that ♦(S) easily implies the existence of an S-club guessing sequence, so one can think of the club-guessing principle as a weak form of diamond. The surprising fact is that very often these club-guessing sequences are provable in ZFC. In [13], Shelah gave various conditions for when certain club-guessing sequences
exist. For example, when S ⊆ Sℵ2 , there is an S-club guessing sequence hC : δ ∈ Si ℵ0 δ
that guesses clubs of ℵ2 in way that for each club E ⊆ ℵ2, there are stationarily many
δ ∈ S such that Cδ ⊆ E. Eisworth and Shelah use certain club-guessing sequences to construct various complicated colorings at successors of singular cardinals. We will prove the existence of analogous club-guessing sequences for weakly inaccessible cardinals, and then use them to construct complicated colorings in the case where the inaccessible cardinal has a stationary subset which does not reflect in an inaccessible cardinal. 15
1.2 Goals For The Paper
Our first goal is to provide a correct proof of the following club-guessing sequence result claimed by Shelah.
THEOREM 1.2.1 (Shelah, [13]) Suppose κ is (weakly) inaccessible, λ < κ is an
κ uncountable regular cardinal, and S is a stationary subset of Sλ . Then there is an
S-club sequence hCδ : δ ∈ Si such that
(i) Cδ is club in δ with order-type λ,
(ii) hcf(α): α ∈ nacc(Cδ)i is strictly increasing with limit δ, and
(iii) for every club E ⊆ κ, there are stationarily many δ ∈ S with Cδ ⊂ E.
After proving Claim 1.2.1, the goal of the paper is to prove our main theorem.
Main Theorem. Suppose
i). κ is an inaccessible cardinal,
ii). ω < λ = cf(λ) < κ, and
κ iii). S ⊆ Sλ is a stationary subset that does not reflect in any inaccessible.
Then there is c :[κ]2 → κ and an ideal I associated with the sequence from Claim
2 ∗ 1.2.1 such that for every unbounded subset A ⊆ κ, we have rng(c [A] ) ∈ I .
The other smaller goals of the paper will serve to prove the main theorem. We will break these goals down by chapter. In Chapter 2, we establish the existence of a nice S-club guessing sequence as in Claim 1.2.1. Also in Chapter 2, we investigate the existence of S-club guessing sequences for Mahlo cardinals κ where S consists of a stationary set of inaccessible cardinals. 16
In Chapter 3, we study weak normality in ideals. Weak normality is a property that generalizes normality, and in some way, is stronger than normality. We determine when weak normality and semi-weak normality coincide in an ideal. We show the behavior of a least function for an ideal affects the weak normality of the ideal. We conclude the chapter by looking at two different methods of creating a weakly normal ideal. In Chapter 4, we begin by formally introducing Todorˇcevi´c’smethod of minimal walks. Then, we discuss an ideal due to Shelah that will be crucial in our proofs: idp(C, I). After characterizing the elements of this ideal, we use a club-guessing sequence from Chapter 2 to define the coloring c that establishes the main theorem.
2 A corollary to the Main Theorem is if κ → [κ]κ, then our ideal is weakly κ-saturated. 17
2 Existence of Nice Club-Guessing Sequences
2.1 Club-Guessing Sequences
The main goal of this chapter is to prove the following theorem:
THEOREM 2.1.1 (Shelah, [13]) Suppose κ is (weakly) inaccessible, λ < κ is an
κ uncountable regular cardinal, and S is a stationary subset of Sλ . Then there is an
S-club sequence hCδ : δ ∈ Si such that
(i) Cδ is club in δ with order-type λ,
(ii) hcf(α): α ∈ nacc(Cδ)i is strictly increasing with limit δ, and
(iii) for every club E ⊆ κ there are stationarily many δ ∈ S with Cδ ⊂ E.
This theorem was stated by Shelah in Cardinal Arithmetic, but the proof given was incorrect. A similar result dealing with successors of singular cardinals was also claimed in Cardinal Arithmetic; the flawed proof in this situation was repaired by Eisworth and Shelah in [7]. We adapt their techniques to give the above result for inaccessible cardinals. Before presenting our proof, we discuss club-guessing sequences in general and outline some of the techniques we will be using. We begin with some vocabulary.
DEFINITION 2.1.1 Suppose C is a club subset of an ordinal δ. Then
(1) acc(C) = {α ∈ C : α = sup(C ∩ α)}, and
(2) nacc(C) = C \ acc(C).
For a set C, acc(C) denotes the set of accumulation points of C and nacc(C) is the set of non-accumulation points. 18
DEFINITION 2.1.2 Let κ be a cardinal and S ⊆ κ be a stationary set of limit
ordinals. We say that a sequence C = hCδ : δ ∈ Si is an S-club system if Cδ is a club subset of δ for each δ ∈ S.
When S = κ, we call the S-club sequence a C-sequence.
DEFINITION 2.1.3 Suppose κ is an uncountable cardinal and S ⊆ κ is a station- ary subset. We say hCδ : δ ∈ Si is an S-club guessing sequence if C is an S-club sequence such that for each club E ⊆ κ, there are stationarily many δ ∈ S such that
Cδ “guesses” E ∩ δ.
There are a few ways an S-club guessing sequence can “guess” a club. When we want to “guess” a club, we typically require one of the following:
1. Cδ ⊆ E,
2. Cδ \ E is bounded in δ,
3. δ = sup(E ∩ nacc(Cδ)), or
4. δ = sup(Cδ ∩ E).
The following concepts are due to Shelah and will be used in our proof. Rather than using their original names, we will use the more descriptive names (due to Kojman in [10]).
DEFINITION 2.1.4 Let C and E be sets of ordinals with E closed. Define
Drop(C,E) = {sup(E ∩ α): α ∈ C \ min(E) + 1}.
If α ∈ nacc(C), then we define Gap(α, C), the gap in C determined by α, by
Gap(α, C) = (sup(C ∩ α), α). 19
If C and E are both subsets of some cardinal κ and heα : α < κi is a C-sequence, then for each α ∈ nacc(C) ∩ acc(E), we define
F ill(α, C, E) = Drop(eα,E) ∩ Gap(α, C).
These operations are used to modify a given S-club sequence in an attempt to transform it into a sequence that guesses clubs. We first lay out a few properties of the Drop operation.
FACT 2.1.1 Suppose κ is a cardinal, E ⊆ κ is a club, δ ∈ acc(E) and C is club in δ. Then
i). Drop(C,E) ⊆ E.
ii). Drop(C,E) is club in δ.
Proof: The proof of these facts is fairly straightforward. The first fact is true because sup(E ∩ α) ∈ E as E is club. For the second fact, we start by showing Drop(C,E) is unbounded in δ. Let γ < δ and choose α ∈ C and β ∈ E such that
γ < β < α.
This can be done because C is club in δ and δ ∈ acc(E). From this, we have
γ < β ≤ sup(E ∩ α).
As sup(E ∩ α) ∈ Drop(C,E), we have Drop(C,E) is unbounded in δ. Next, we show Drop(C,E) is closed. Suppose ξ < cf(δ) and
haα : α < ξi ⊂ Drop(C,E) is a strictly increasing sequence with
a = sup{aα : α < ξ}. 20
For each α < ξ, there is bα ∈ C such that
aα = sup(E ∩ bα).
Since C is closed in δ, we have b = sup{bα : α < ξ} ∈ C. From this, we get a ≤ sup(E ∩ b). We actually have equality because if a < sup(E ∩ b), then there would be α < ξ such that a < aα. So Drop(C,E) is club in δ.
The following theorem by Shelah is one of the first results for club-guessing se- quences. This proof is a prototype of most results relating to the existence of a club-guessing sequence. The general idea behind the proof is if there is no club- guessing sequence with the desired properties, then we correct the sequence using the Drop. Now, if this new sequence does not meet the requirements, then we correct the new sequence using the Drop. Then, we argue that eventually one of these corrected sequences has the required properties.
THEOREM 2.1.2 (Shelah, [13]) Suppose λ < κ are regular cardinals with λ+ <
κ κ, and S ⊆ Sλ is stationary. Then there exists an S-club guessing sequence hCδ : δ ∈ Si such that for any club E ⊆ κ, the set
{δ ∈ S : Cδ ⊂ E} is stationary.
Proof: Assume the contrary and suppose there is no sequence hCδ : δ ∈ Si with the
+ desired property. We will construct a sequence hEξ : ξ < λ i of clubs of κ and a ξ sequence of club-guessing sequences C to arrive at a contradiction. To start with,
0 0 let E0 = κ and C be a sequence of subsets of κ such that Cδ ⊂ δ and otp(Cδ) = λ for each δ ∈ S. ζ When ξ = ζ + 1, C does not have the property we are looking for. So, there is a
0 club Eζ ⊆ κ such that ζ 0 {δ ∈ S : Cδ ⊂ Eζ } 21
is non-stationary, or
0 ζ 0 δ ∈ S ∩ Eζ ⇒ Cδ 6⊂ Eζ .
0 ζ We define Eξ = acc(Eζ ∩ Eζ ). To define C , we will use the Drop operation. If
δ ∈ acc(Eξ) ∩ S, then we let
ξ Cδ = Drop(Cδ,Eξ).
ξ If δ ∈ nacc(Eξ) ∩ S, then let Cδ = Cδ. When ξ is a limit ordinal, we let
\ Eξ = Eζ . ζ<ξ
ξ And, we modify C just like above. Let \ E = acc Eξ , ξ<λ+ + and let δ ∈ S ∩ E. Since δ ∈ S ∩ E, we know δ ∈ S ∩ Eξ for all ξ < λ , which means
ξ Cδ 6⊂ Eξ
+ + ξ for all ξ < λ . As δ ∈ acc(Eξ) for all ξ < λ , it follows that Cδ = Drop(Cδ,Eξ).
+ For each α ∈ Cδ, there is ξα < λ such that for any ξ ≥ ξα,
sup(Eξα ∩ α) = sup(Eξ ∩ α).
+ This follows from the fact that hsup(Eξ ∩ α): ξ < λ i is a non-increasing sequence
+ of ordinals and must be eventually constant. Since otp(Cδ) = λ < λ , we know
∗ + ξ = sup{ξα : α ∈ Cδ} + 1 < λ .
By construction, for any α ∈ Cδ, we have
sup(Eξ∗ ∩ α) = sup(Eξ∗+1 ∩ α) ∈ Eξ∗+1. 22
This in turn shows that
ξ∗ ξ∗ Cδ = Drop(Cδ ,Eξ∗ ) ⊂ Eξ∗+1,
contradicting the choice of Eξ∗+1.
Theorem 2.1.2 establishes the existence of a club-guessing sequence on a lot of cardinals, which is nice, but there is room for improvement. When κ is the successor of a singular cardinal, Theorem 2.1.2 can be improved, which is stated as Claim 2.6 in [13]. The new club-guessing sequence still guesses clubs in the previous fashion with a stronger condition on the non-accumulation points.
THEOREM 2.1.3 (Shelah, claimed in [13], correct proof in [7]) Suppose κ =
+ κ µ , ω < λ = cf(µ) < µ, and S ⊆ Sλ is stationary. Then there is a sequence hCδ : δ ∈ Si such that
(i). otp(Cδ) = λ,
(ii). hcf(α): α ∈ nacc(Cδ)i is a strictly increasing sequence in δ, and
(iii). for every club E ⊆ κ, there are stationarily many δ ∈ S such that Cδ ⊂ E.
Club-guessing sequences from with properties (i)-(iii) are called “nice club-guessing sequences.” Theorem 2.1.3 shows that nice club-guessing sequences exist when κ is the successor of a singular cardinal of uncountable cofinality. As mentioned earlier, we give a correct version of Theorem 2.1.3 in the context of inaccessible cardinals. Shelah also investigates club-guessing sequences at Mahlo cardinals, taking a look at the existence of S-club guessing sequences when S is a stationary set of inaccessible cardinals. The results he obtains are stated without proof, and it is not clear if they are correct. In the last section of this chapter, we will achieve a partial rescue of one of his claims. 23
2.2 Nice Club-Guessing Sequences: Uncountable Cofinality
In this section, we prove Theorem 1.2.1 from the introduction. Our first step is to prove the following lemma:
LEMMA 2.2.1 Suppose κ is an inaccessible cardinal, λ < κ is an uncountable reg-
κ ular cardinal, and S ⊆ Sλ is stationary. Then there exists an S-club sequence such that:
(i). |Cδ| < δ, and
(ii). for every club E ⊂ κ, there are stationarily many δ ∈ S such that for all τ < δ,
{α ∈ nacc(Cδ) ∩ E : cf(α) > τ} is unbounded in δ.
κ Proof: Let S ⊆ Sλ is stationary. By way of contradiction, assume that there is no
S-club sequence with property (ii). Let hCδ : δ ∈ Si be any S-club sequence such
that |Cδ| < δ. We will construct by recursion on n < ω
n n • C := hCδ : δ ∈ Si, an S-club sequence,
n • τδ < δ, a regular cardinal,
n • δ < δ, and
• En ⊆ κ, a club in κ.
0 0 0 n n Let C = hCδ : δ ∈ Si, E0 = κ, δ = 0 and τδ = 0. Suppose the objects C , En, δ
n and τδ have been constructed. Now we describe the n + 1 stage of the construction.
0 By our assumption, there is a club En ⊆ κ such that
n 0 {δ ∈ S : ∀τ < δ({α ∈ nacc(Cδ ) ∩ En : cf(α) > τ} is unbounded)}
1 is non-stationary. So, there is another club En ⊆ κ such that
1 n 0 En ∩ {δ ∈ S : ∀τ < δ({α ∈ nacc(Cδ ) ∩ En : cf(α) > τ} is unbounded)} = ∅ 24
1 or if δ ∈ En ∩ S, then there are τ, < δ such that
n 0 α ∈ nacc(Cδ ) ∩ En \ ( + 1) ⇒ cf(α) ≤ τ.
0 1 n+1 n+1 Define En+1 = acc(En ∩ En ∩ En), and δ , τδ < δ be least and τ corresponding to the above comment.
We say δ is active at stage n + 1 if δ ∈ acc(En+1). If δ ∈ S is inactive at stage
n+1 n n+1 n n+1 n n + 1, we let Cδ = Cδ , δ = δ and τδ = τδ . Suppose δ is active at stage n + 1. We say α < δ needs attention at stage n + 1 if
n n+1 α ∈ nacc(Cδ ) ∩ acc(En+1) \ δ + 1.
n+1 Any ordinal α < δ that needs attention at stage n + 1 has cofinality less than τδ by our earlier comment.
n+1 n n n To construct Cδ , we first we let Dδ = Drop(Cδ ,En+1). The set Dδ is club in
n δ as δ ∈ acc(En+1) and Cδ is club in δ. If α needs attention at stage n + 1, then
n α ∈ acc(En+1) and α = sup(α ∩ En+1). We also have α ∈ nacc(Dδ ), for if this isn’t
n the case, then we get a contradiction to fact that α ∈ nacc(Cδ ). This fact allows us
n to conclude that Fill(α, Dδ ,En+1) is defined for any α that needs attention at stage n + 1. Now, we define
n+1 n [ n Cδ = Dδ ∪ {Fill(α, Dδ ,En+1): α needs attention)}.
n+1 n n+1 n+1 The set Cδ is unbounded in δ because Dδ ⊂ Cδ , and it is closed because Cδ
n n+1 was built by adding closed sets into the gaps of Dδ . We also have |Cδ | < δ because
n+1 n n+1 n |Cδ | ≤ |Cδ | + τδ · |Cδ | < δ.
n n n The previous inequality holds for |Dδ | ≤ |Cδ | < δ, we added in at most |Dδ | many n+1 n+1 sequences of length at most τδ . C has been constructed to fulfill our needs, and the recursion can go on. 25
T Let E = n<ω En. For each δ ∈ acc(E) ∩ S, we define
n δ = sup{δ : n < ω} and
n θδ = sup{|Cδ | : n < ω}.
n Each δ and θδ is strictly less than δ because ω < cf(δ) and |Cδ | < δ. Next, we let
ηδ = max{δ, θδ} + 1,
and we have ηδ < δ. Using Fodor’s theorem, we can pass to a stationary subset
1 ∗ 1 S ⊆ S ∩ acc(E) such that ηδ = η for each δ ∈ S .
1 ∗ + Now, we can choose δ ∈ S such that |η | divides otp(E ∩ δ). As E ⊂ acc(En) for all n, we have δ is active at all stages of the construction. Since δ ∈ acc(E) and |η∗|+ divides otp(E ∩ δ), |E ∩ δ \ η∗| ≥ |η∗|+.
As |η∗|+ > η∗, we can find an ordinal γ such that
• γ ∈ E
• η∗ < γ < δ and
S n • γ 6∈ n<ω Cδ .
Now, we construct the sequence hαn : n < ωi of ordinals by
n αn = min(Cδ \ γ), and prove this sequence is strictly decreasing.
S n By definition, αn is strictly greater than γ because γ 6∈ n<ω Cδ . This means
n n+1 ∗ αn is an element of nacc(Cδ ). From the definition of αn, we know δ < η < αn.
There are two cases to consider with αn: either αn needs attention at stage n+1 or not. 26
Case 1: αn does not need attention at stage n + 1.
Since αn does not need attention at stage n + 1, we have αn 6∈ acc(En+1), and by letting βn = sup(En+1 ∩ αn) < αn, we have, as γ ∈ E ⊂ En+1, γ ≤ βn < αn. Also, βn
n n+1 is an element of Dδ ⊂ Cδ , so αn+1 ≤ βn < αn.
Case 2: αn needs attention at stage n + 1.
n In this situation, αn ∈ nacc(Cδ)∩acc(En+1), and Fill(αn,Dδ ,En+1) is defined and club in αn. Since γ < αn, we have
n γ < αn+1 ≤ min(Fill(αn,Dδ ,En+1)) < αn
and αn+1 < αn. In both cases, we have αn+1 < αn, and hαn : n < ωi is a strictly decreasing sequence of ordinals, a contradiction. Hence, there is an S-club guessing sequence with the desired properties.
Even though the bound on this sequence is not uniform, we can prove there is a sequence with a uniform bound on all the terms and have the same guessing property. In the upcoming proofs, we will be working with property (ii) quite a bit, so we will introduce some ad-hoc terminology. Let E be a club of κ. We say C guesses E at δ if
{α ∈ E ∩ nacc(Cδ): cf(α) > τ} is unbounded in δ for each τ < δ.
COROLLARY 2.2.1 Suppose κ is an inaccessible cardinal, λ < κ is an uncountable
κ regular cardinal, and S ⊆ Sλ is stationary. There is a C-sequence C = hCδ : δ ∈ Si and µ < κ such that
(i). |Cδ| < µ, and 27
(ii). for every club E ⊆ κ, there are stationarily many δ ∈ S such that for all τ < δ,
{α ∈ nacc(Cδ) ∩ E : cf(α) > τ} is unbounded in δ.
Proof: Let C be the S-club guessing sequence from Lemma 2.2.1. Assume the contrary. Suppose for each µ < κ, there is a club Eµ ⊆ κ such that if C guesses Eµ at δ, then
|Cδ| ≥ µ.
Now, let E = {δ < κ :(∀α < δ)(δ ∈ Eα)} which is club in κ. Since there are stationarily many places where C guesses E, we can pick δ ∈ E where C guesses E
at δ. Since C guesses E at δ, it follows that C guesses Eα at δ for each α < δ. This in turn implies that
|Cδ| ≥ α
for all α < δ, or
|Cδ| ≥ δ.
This contradicts (i) of Lemma 2.2.1, and the corollary is proved.
The next lemma shows the S-club sequence from Lemma 2.2.1 implies the exis- tence of a nice S-club guessing sequence.
LEMMA 2.2.2 Suppose κ is an inaccessible cardinal, λ < κ is an uncountable reg-
κ ular cardinal, and S ⊆ Sλ is a stationary subset. Then the following are equivalent:
(1). There is an S-club system hCδ : δ ∈ Si such that
(i). |Cδ| < δ, and
(ii) for every club E ⊆ κ, there are stationarily many δ ∈ S such that for all
τ < δ, {α ∈ nacc(Cδ) ∩ E : cf(α) > τ} is unbounded in δ.
(2). There is an S-club system hCδ : δ ∈ Si and µ < κ such that 28
(i). |Cδ| < µ, and
(ii) for every club E ⊆ κ, there are stationarily many δ ∈ S such that for all
τ < δ, {α ∈ nacc(Cδ) ∩ E : cf(α) > τ} is unbounded in δ.
(3). There is an S-club system hCδ : δ ∈ Si such that
(i). otp(Cδ) = λ,
(ii). hcf(α): α ∈ nacc(Cδ)i is strictly increasing sequence with limit δ, and
(iii). for every club E ⊆ κ, there are stationarily many δ ∈ S such that Cδ ⊂ E.
Proof: For (1) ⇒ (2), this is just Corollary 2.2.1. Now, we show (2) ⇒ (3). Our proof and notation is similar to the proof of Theorem 1 in [2]. To start, we prove if C is as in (2), then there is a club E∗ ⊆ κ such that for every club E ⊆ κ, there are stationarily many δ ∈ S where C guesses
∗ ∗ acc(E ) at δ and Drop(Cδ,E ) ⊂ E Suppose this is not the case. By recursion, we will construct a µ-sequence of clubs
Eξ of κ as follows:
Case 1: ξ = 0: We start with E0 = κ,
T Case 2: ξ is a limit: We let Eξ = ζ<ξ Eζ .
∗ Case 3: ξ = ζ + 1: From our assumption, Eζ cannot be our E . This means we
0 1 0 can find clubs Eζ and Eζ such that for all δ ∈ Eζ , if C guesses acc(Eζ ) at δ, then
1 Drop(Cδ,Eζ ) 6⊆ Eζ ,
or there is α ∈ Cδ \ min(Eζ ) + 1 such that
1 sup(Eζ ∩ α) 6∈ Eζ . 29
0 1 We define the next club Eξ by Eξ = acc(Eζ ∩ Eζ ∩ Eζ ). T Now, let E = ξ<µ Eξ, which is club in κ, so by (ii) of (2), we can find δ ∈ S where C guesses E at δ and δ > µ. Since C guesses E at δ, it follows that δ ∈ E, and from
0 the definition of E, we know δ ∈ Eξ for each ξ < µ. Also, we have C guesses acc(Eξ)
at δ because E ⊆ acc(Eξ). Hence, for each ξ < µ, there is α ∈ Cδ \ min(Eξ) + 1 with
1 sup(Eξ ∩ α) 6∈ Eξ ⊇ Eξ+1.
As hEξ : ξ < µi is a decreasing sequence of clubs, for each α ∈ Cδ \ min(E) + 1, it
follows that hsup(Eξ ∩ α): ξ < µi is non-increasing sequence and must be eventually
constant. For each α ∈ Cδ, there are γα < δ and ξα < µ such that if ξα ≤ ξ, then
sup(Eξ ∩ α) = γα.
Since |Cδ| < µ, it follows that
∗ ξ = sup{ξα : α ∈ Cδ} < µ.
Since C guesses acc(Eξ∗ ) at δ, there is α ∈ Cδ \ min(Eξ∗ ) + 1 such that
sup(Eξ∗ ∩ α) 6∈ Eξ∗+1.
∗ Yet ξ ≥ ξα, and we have
sup(Eξ∗ ∩ α) = γα = sup(Eξ∗+1 ∩ α) ∈ Eξ∗+1,
a contradiction. With the existence of such E∗, we can continue our proof. If C guesses acc(E∗)
at δ, we can construct Dδ by recursion on λ. Before we start, for each δ for which C
∗ δ guesses acc(E ), fix an increasing λ-sequence hτξ : ξ < λi which is cofinal in δ.
δ First, for ξ = 0, let α0 = 0. For ξ = ζ + 1, choose
δ ∗ δ αξ ∈ {α ∈ nacc(Cδ) ∩ acc(E ): cf(α) > τξ } 30
δ δ ∗ such that αζ < αξ. This is possible since C guesses acc(E ) at δ. When ξ is a limit,
we let αξ = sup{αζ : ζ < ξ}, and our construction can continue. In the end, we define
Dδ = {αξ : ξ < λ}.
∗ From our method of construction, when C guesses acc(E ) at δ, Dδ has the fol- lowing properties:
∗ (i). Dδ ⊆ Drop(Cδ,E )
(ii). Dδ is club in δ with otp(Dδ) = cf(δ), and
(iii). hcf(α): α ∈ nacc(Dδ)i is increasing and cofinal in δ.
∗ ∗ Fact (i) holds because Dδ ⊂ Cδ ∩ acc(E ). If α ∈ Dδ, then sup(E ∩ α) = α and
α ∈ Cδ. Fact (ii) and (iii) hold by how we constructed Dδ. For other δ ∈ S, we can pick Dδ to be a subset of δ satisfying (ii) and (iii). It is easy to see hDδ : δ ∈ Si is an S-club system meeting the requirements of (3), and (2) ⇒ (3). It is clear to see that (3) ⇒ (1).
Recall, the goal of this section was to prove the following theorem:
Theorem 2.1.1: Suppose κ is (weakly) inaccessible, λ < κ is an uncountable regular
κ cardinal, and S is a stationary subset of Sλ . Then there is an S-club sequence hCδ : δ ∈ Si such that
(i) Cδ is club in δ with order-type λ,
(ii) hcf(α): α ∈ nacc(Cδ)i is strictly increasing with limit δ, and
(iii) for every club E ⊆ κ, there are stationarily many δ ∈ S with Cδ ⊂ E.
Combining Lemma 2.2.2 and Corollary 2.2.1, we get the existence of a nice S-
κ club sequence on an inaccessible cardinal κ when S ⊆ Sλ for a regular cardinal λ of uncountable cofinality. This establishes Theorem 2.1.1. 31
Finding a nice S-club guessing sequences when S is a stationary subset of ordinals with countable confinality is a bit tricky. Club subsets of an ordinal with countable cofinality only need to be unbounded, and this fact causes trouble. It is an open question whether nice S-club guessing sequences exist when S is a stationary subset consisting of ordinals with countable cofinality. In [7], Eisworth and Shelah constructed a “well-formed” S-club sequence to replace a nice S-club guessing sequence. At a first glance, it seems possible for there to be a well-formed S-club guessing sequence when S is a stationary subset consisting of ordinals of countable cofinality. The S-club guessing results obtained in this section used arguments very similar to those used in the case of successors of a singular cardinals. The next step would be to find conditions for the existence of a well- formed S-club guessing sequence when S consists of ordinals of countable cofinality. This leads to the following questions.
κ QUESTION 2.2.1 Suppose κ is an inaccessible cardinal and S ⊆ Sω is a stationary subset of κ. Does there exist a well-formed S-club guessing sequence?
A positive answer would be a first step towards a partition theorem when S consists of ordinals of countable cofinality.
2.3 Constructing A Club-Guessing Sequence Using ⊗C
In this section, we will focus on constructing a club-guessing sequence when the stationary subset consists of regular cardinals. The proof uses a club-sequence having the following property:
⊗C : For every club E ⊆ κ, there are stationarily many δ ∈ S such that E ∩ δ \ Cδ is unbounded in δ 32
Under what situations is there an S-club sequence satisfying ⊗C ? Suppose λ <
κ κ are regular cardinals, S ⊆ Sλ is stationary, and C is an S-club sequence with
otp(Cδ) = λ for each δ ∈ S. Then ⊗C holds. Why? If E ⊆ κ is a club, then
{δ < κ : otp(E ∩ δ) = δ}
is club in κ. Now, for any δ ∈ S ∩ {δ < κ : otp(E ∩ δ) = δ}, we have otp(E ∩ δ) = δ
and otp(Cδ) = λ < δ. Since otp(Cδ) = λ < δ = otp(E ∩ δ), it follows that E ∩ δ \ Cδ
is unbounded in δ. Hence, ⊗C holds.
What makes ⊗C an interesting property is that it is SOMETIMES possible to
find an S-club sequence satisfying ⊗C when S is a stationary subset of inaccessible cardinals:
CLAIM 2.3.1 (Shelah, [13]) Suppose κ is an inaccessible cardinal and there is a sequence hSi : i < κi of stationary subsets of κ such that for each δ < κ, Si ∩ δ is not stationary in δ for some i < δ. Then for every stationary S ⊆ κ consisting of
inaccessible cardinals, there is an S-club system satisfying ⊗C .
Along with conditions for the existence of an S-club sequence satisfying ⊗C , Shelah
showed the existence of an S-club sequence on κ satisfying ⊗C implies κ cannot be weakly compact. This was done by way of constructing a function c :[κ]2 → ω that has an infinite range on every unbounded subset of κ (see Claim 4.9 in [13]).
The property ⊗C was introduced in Claim 2.12 of Chapter III of [13]. Claim 2.12 states the existence of a nice S-club guessing sequence from ⊗C and conditions for
this modified club-guessing sequence to satisfy ⊗C , but there is a problem with Claim 2.12. We give a partial rescue to Claim 2.12 with Lemma 2.3.1, but before proving the lemma, we need two facts about Drop.
FACT 2.3.1 Suppose E ⊆ κ is club, δ ∈ acc(E) and Cδ is club in δ. If β ∈
acc(Drop(Cδ,E)), then β ∈ Cδ. 33
Proof: Given β ∈ acc(Drop(Cδ,E)), there is an increasing sequence hβi : i <
cf(β)i ⊂ Drop(Cδ,E) converging to β. As βi ∈ Drop(Cδ,E), there is αi ∈ Cδ such that
βi = sup(E ∩ αi)
for each i < cf(β). Since hβi : i < cf(β)i is increasing and converges to β, it follows
that hαi : i < cf(β)i is increasing and converges to β. As Cδ is club in δ and αi → β, we have β ∈ C.
Using Fact 2.3.1, we get a slightly more useful fact.
FACT 2.3.2 Suppose E ⊆ κ is a club, δ ∈ acc(E), Cδ is club in δ. If β ∈
Drop(Cδ,E) and β 6∈ Cδ, then β ∈ nacc(Drop(Cδ,E)).
Proof: Since β ∈ Drop(Cδ,E), we know either β ∈ nacc(Drop(Cδ,E)) or β ∈
acc(Drop(Cδ,E)). From Fact 2.3.1 and knowing β 6∈ Cδ, the only possibility is
β ∈ nacc(Drop(Cδ,E)).
Now, we can prove the lemma.
LEMMA 2.3.1 (Shelah, [13]) Suppose κ is an inaccessible cardinal, S is a station- ary subset of {α < κ : α is an inaccessible}, and there is an S-club system C that sat-
∗ ∗ isfies ⊗C . There is a club E ⊆ κ such that for hDδ : δ ∈ Si with Dδ = Drop(Cδ,E ) has the following property
(1) For every club E ⊂ κ, there are stationarily many δ ∈ S such that
δ = sup(E ∩ nacc(Dδ)).
Proof: Assume the contrary, and suppose for every club E of κ, property (1) fails. This means for each club E ⊆ κ, there is a club F of κ such that
δ 6= sup(F ∩ nacc(Drop(Cδ,E))) 34
for every δ ∈ S. Now, we will construct an ω-sequence, hEn : n < ωi, of clubs of κ
δ and βn for each δ ∈ S.
δ For n = 0, we define E0 = λ and β0 = 0 for each δ ∈ S.
δ When n = k + 1, suppose Ek and βk for each δ ∈ S have been defined. Since
0 property (1) fails to hold for any club of κ, there is a club Ek ⊆ κ such that for any δ ∈ S, we have
0 δ 6= sup(Ek ∩ nacc(Drop(Cδ,Ek))).
δ For each δ ∈ S, there is γk < δ such that
0 δ sup(Ek ∩ nacc(Drop(Cδ,Ek))) = γk.
Next, we let
δ δ δ βk+1 = max{γk, βk} + 1 and
0 Ek+1 = acc Ek ∩ Ek , and the construction continues. T Define E = n<ω En, which is club in κ. By ⊗C , there is δ ∈ S such that E ∩δ\Cδ is unbounded in δ. As E ⊆ En for each n < ω, it follows that En ∩ δ \ Cδ is also unbounded in δ. Let
δ δ β = sup{βn : n < ω},
∗ which is less than δ because δ is an inaccessible cardinal. Choose β ∈ E ∩ δ \ Cδ greater than βδ. Before we can arrive at a contradiction, we first need the following claim.
∗ CLAIM 2.3.2 For each n < ω, β 6∈ Drop(Cδ,En).
∗ Proof: Assume the contrary and suppose β ∈ Drop(Cδ,En) for some n < ω. As
∗ ∗ β ∈ E, we know β ∈ En+1, which also gives us that
∗ β ∈ En+1 ∩ Drop(Cδ,En). 35
∗ δ ∗ Since β > β and β ∈ En+1 ∩ Drop(Cδ,En), we have
∗ β ∈ acc(Drop(Cδ,En)).
∗ Yet, we know β ∈ En ∩ δ \ Cδ, so it follows from Fact 2.3.2 that
∗ β ∈ nacc(Drop(Cδ,En)).
∗ δ ∗ A contradiction to the fact that β > β . Hence, β 6∈ Drop(Cδ,En) for all n < ω, and our claim is proved. Now, we will construct a strictly decreasing sequence of ordinals and obtain a contradiction. For each n < ω, we let
∗ αn = min(Drop(Cδ,En) \ β ).
For each n < ω, we have
αn ∈ nacc(Drop(Cδ,En)) and
∗ δ αn > β > βn.
From these two facts, it follows that αn 6∈ En+1, and we let
γn = sup(En+1 ∩ αn),
which is strictly less than αn. Since
∗ ∗ β ∈ E ⊆ Ek for all k < ω, we have
∗ β ≤ γn < αn.
Let βn ∈ Cδ witness αn = sup(En ∩ βn). From the fact that αn = sup(En ∩ βn), we know
(αn, βn) ∩ En = ∅, 36
and since En+1 ⊆ En, we get
(αn, βn) ∩ En+1 = ∅.
Also, from the definition of γn, we know
(γn, αn) ∩ En+1 = ∅.
Combing the previous facts, we get
γn = sup(En+1 ∩ βn) ∈ Drop(Cδ,En+1).
Hence,
∗ αn+1 = min(Drop(Cδ,En+1) \ β ) ≤ γn < αn.
Thus, the sequence hαn : n < ωi is strictly decreasing, a contradiction. Therefore, there is a club E∗ ⊆ κ with the desired properties.
The purpose of Claim 2.12 in [13] was to prove the existence of a nice S-club guessing sequence satisfying ⊗C . The proof of Claim 2.12 was sketched out, but there is a flaw in the sketch. Shelah fills in the troublesome non-accumulation points
using glσ which does not have the necessary properties. This leads to the following question.
QUESTION 2.3.1 Suppose κ is an inaccessible cardinal and S is a stationary subset consisting of inaccessible cardinals. Also, suppose C is an S-club sequence satisfying
⊗C . Is there a nice S-club guessing sequence that satisfies ⊗C ?
Lemma 2.3.1 was the first step in trying to answer this question. In the successors of a singular cardinals versions of Lemma 2.2.1 and Lemma 2.2.2, the proofs were fixed by filling in the ordinals that “needed attention,” so it seems that it may be possible to salvage Claim 2.12. The existence of a nice S-club guessing sequence would be a step in the direction of constructing a coloring when working with a stationary subset of inaccessible cardinals. 37
3 Weakly Normal Ideals
3.1 An Introduction to Ideals and Filters
In this chapter, we will look at the effects of weak normality on an ideal. We begin with some basics, and then we introduce weak normality and highlight some properties equivalent to it. After this, we show a filter is weakly normal if and only if the filter is semi-weakly normal and weakly κ-saturated. This was first proved by
Yoshihiro Abe for filters on Pκ(λ); we show that essentially the same proof works in the case of filters on κ, and furthermore we eliminate the need for his assumption that the filters are κ-complete. We connect these ideas to the existence of least functions for an ideal. We start with the basics of filters and ideals.
DEFINITION 3.1.1 A filter on a non-empty set S is a collection F of subset of S such that
i. S ∈ F and ∅ 6∈ F ,
ii. if X ∈ F and Y ∈ F , then X ∩ Y ∈ F ,
iii. if X,Y ⊂ S, X ∈ F and X ⊂ Y , then Y ∈ F .
An ideal on a non-empty set S is a collection I of subsets of S such that
i. ∅ ∈ I and S 6∈ I,
ii. if X ∈ I and Y ∈ I, then X ∪ Y ∈ I,
iii. if X,Y ⊂ S, X ∈ I and Y ⊂ X, then Y ∈ I. 38
If C is a collection of subsets of a non-emtpy set S, we define the dual of C to be the collection {S \ X : X ∈ C}.
We denote the dual of C by C∗. If F is a filter on S, then F ∗ is an ideal on S. Likewise, if I is an ideal on S, the I∗ is a filter on S. For an ideal I on a set X, a set A ⊆ X is I-positive if A 6∈ I, and the collection of all I-positive sets is denoted by I+. It is obvious that a set A is I-positive if and only if A ∩ X 6= ∅ for every X ∈ I∗. A maximal filter is called an ultrafilter. If F is an ultrafilter on a set X, then either X \ A ∈ F or A ∈ F . A filter F is uniform if |X| = κ for all X ∈ F , and an ideal is uniform if the dual of I is uniform. We will also need to discuss the degree of completeness of a filter. Every filter is closed under finite intersections, but not necessarily infinite intersections. There is a special name for filters that are closed under infinite intersections.
DEFINITION 3.1.2 Suppose F is a filter and κ ≥ ω is a cardinal. The filter F is
called κ-complete if for every γ < κ and hAi : i < γi ⊂ F , we have
\ Ai ∈ F. i<γ An ideal I is κ-complete if the dual of I is κ-complete.
One important filter is the club filter. For an uncountable regular cardinal κ, the club filter is the collection
{X ⊆ κ : ∃C ⊆ X(C is club in κ)},
and we will denote the club filter on κ by C(κ). The dual of the club filter is called the non-stationary ideal and is denoted by NS(κ). If κ is an uncountable regular cardinal, then the club filter is κ-complete. 39
Unless stated, we are going to assume our ideals extend the non-stationary ideal. This implies that the ideals will be uniform.
3.2 Introduction to Weak Normality
In this chapter, we will look at some results concerning weak normality. Weakly normal filters were first introduced by Kanamori in [9] as a generalization of normal ultrafilters on measurable cardinals. We begin with some definitions.
DEFINITION 3.2.1 Let κ be a cardinal and I an ideal on κ.
i. A function f : κ → κ is unbounded if {δ < κ : γ < f(δ)} ∈ I+ for all γ < κ.
ii. A function f : κ → κ is a least function if f is unbounded, and if g : κ → κ is a function such that
{δ < κ : g(δ) < f(δ)} ∈ I∗,
then there is a γ < κ such that
{δ < κ : g(δ) ≤ γ} ∈ I∗.
iii. F is a p-point filter if for every unbounded function f : κ → κ, there is a set X ∈ I+ such that for every γ < κ,
|f −1({γ}) ∩ X| < κ.
DEFINITION 3.2.2 A filter F is normal if F is closed under diagonal intersec- tions. In other words, if hAi : i < κi is a sequence of elements of F , then
∆i<κAi = {δ < κ : ∀i < δ (δ ∈ Ai)} ∈ F.
We say an ideal I is normal if I∗ is a normal filter. 40
One useful property of a normal filter is every regressive function defined on an I-positive set is constant on an I-positive set, and it is this property weak normality attempts to generalize. Weak normality eases up on the behavior of the function on the set in question by letting the function be bounded instead of constant, but it requires the restricted set to be larger.
DEFINITION 3.2.3 An ideal I on a cardinal κ is weakly normal if for every regressive function h : κ → κ there is γ < κ such that
{α < κ : h(α) ≤ γ} ∈ I∗.
This is the same as saying the identity function is a least function.
Weak normality is in a sense both stronger and weaker than normality. Weak normality is stronger in the sense that every regressive function is bounded on a large set, but it is also weaker because we get a set where the function is bounded instead of constant. If we weaken the conditions on weak normality, we arrive at the notion of semi-weak normality.
DEFINITION 3.2.4 Suppose I is an ideal on a cardinal κ. The ideal I is semi- weakly normal if for every X ∈ I+ and regressive function f : X → κ there is γ < κ such that {α ∈ X : f(α) ≤ γ} ∈ I+.
Semi-weak normality takes a step back from weak normality by only requiring the function to be bounded on an I-positive set. There will be times where semi- weak normality and weak normality are equivalent, and there will be situations where weak normality and normality are equivalent. But first, we look at characterizations of weak normality due to Kanamori.
PROPOSITION 3.2.1 (Kanamori, [9]) For any filter F over κ, the following are equivalent: 41
(i). F is weakly normal.
(ii) Every filter extension of F is weakly normal.
+ (iii) If hXα : α < κi ⊆ F such that Xβ ⊆ Xα for α < β, then
+ ∆β<κXβ = {β < κ : α < β → β ∈ Xα} ∈ F .
(iv) F is a p-point filter extending the club filter on κ.
Characterization (iii) is very similar to the definition of normality, which hints at a possibility of conditions on an ideal where weak normality and normality are equivalent. As it turns out, these two concepts are same when the ideal is sufficiently closed.
PROPOSITION 3.2.2 (Kanamori, [9]) For any κ-complete filter F over κ, the following are equivalent:
i. F is weakly normal.
ii. F is a normal, κ-saturated filter.
+ An ideal I is κ-saturated if for any {Xα : α < λ} ⊂ I , there is α < β < κ such
+ that Xα ∩ Xβ ∈ I . Proposition 3.2.2 is not useful for us as the requirement that the filter is κ-complete is too strong. Later on, we will determine other conditions connecting normality and weak normality. Also, we will look at other conditions where weak normality and semi-weak normality coincide.
3.3 Weak Normality Arises
In this section, we will look at ideals with characteristics that produce weak nor- mality. As we will see, least functions and how they behave will affect the structure of the ideal. 42
Our first lemma is a folklore result relating to weak normality. The lemma shows that the image of an ideal under a least function will be weakly normal.
LEMMA 3.3.1 Suppose I is an ideal on κ, and there is a least function f : κ → κ for I. Then
−1 f∗(I) = {A ⊆ κ : f (A) ∈ I}
is a weakly normal ideal.
Proof: It is clear that J = f∗(I) is an ideal. All that needs to be shown is that J is weakly normal. Suppose h : κ → κ is a regressive function. The goal is to find γ < κ such that {α < κ : h(α) ≤ γ} ∈ J ∗.
First, define a function g : κ → κ by
g(α) = h(f(α)).
As h is a regressive function, it follows that for all α < κ,
g(α) = h(f(α)) < f(α),
or {δ < κ : g(δ) < f(δ)} ∈ I∗.
Since f is a least function, there is γ∗ < κ such that
{α < κ : g(α) ≤ γ∗} = {α < κ : h(f(α)) ≤ γ∗} ∈ I∗.
Define Y = {β < κ : h(β) ≤ γ∗}, and we need to show that
f −1(Y ) ∈ I∗.
Notice, if β ∈ {α < κ : h(f(α)) ≤ γ∗}, then f(β) ∈ Y , and as f(β) ∈ Y , it follows that β ∈ f −1(Y ). So, we have that
{α < κ : h(f(α)) ≤ γ∗} ⊂ f −1(Y ) 43
and {α < κ : h(f(α)) ≤ γ∗} ∈ I∗, and hence,
f −1(Y ) ∈ I∗.
Thus, J is a weakly normal ideal.
The above is one way to obtain a weakly normal ideal from a least function. As we’ll see later, there is another way to create a weakly normal ideal using a least function. In the meantime, we focus our efforts on determining conditions that create weakly normal ideals. Clearly, a weakly normal ideal is semi-weakly normal, so semi-weakly normal ideals would be the first place to start looking. Before we look at semi-weakly normal ideals, we need to introduce a weak form of saturation.
DEFINITION 3.3.1 Let I be an ideal on a cardinal κ, and let θ be a cardinal. The ideal I is weakly θ-saturated if there is no partition of κ into θ disjoint I-positive sets.
Weak saturation has been studied in [3], where it was used to investigate parti- tion relations in the realm of the successor to a singular cardinal. The ideals used in [3] were not only weakly saturated, but also a weak completion property called indecomposability. In [4], Eisworth showed that under certain conditions, it was impossible for an ideal to have weak saturation and indecomposability. The following proposition is our version of a result first proved by Abe [1] in the
context of κ-complete filters on Pκ(λ). We note that our proof does not require the completeness assumption on the ideal.
PROPOSITION 3.3.1 (Abe, [1]) Suppose κ is a cardinal and I is an ideal on κ. Then I is weakly normal if and only if I is semi-weakly normal and weakly κ-saturated. 44
Proof: Suppose I is weakly normal. Let X ∈ I+ and f : X → κ be a regressive function. First, we extend f to g : κ → κ such that g is regressive. By the weak normality of I, there is γ < κ such that
Y := {α < κ : g(α) ≤ γ} ∈ I∗.
Define Z = X ∩ Y . Since Y ∈ I∗ and X ∈ I+, we know Z ∈ I+. Notice for any α ∈ Z, we have f(α) = g(α) ≤ γ, so it follows that Z ⊂ {α ∈ X : f(α) ≤ γ}.
And the semi-weak normality of I has been established.
Next, we show I is weakly κ-saturated. Assume the contrary, and let hXα : α < κi be a partition of κ into κ-many I-positive sets. We may assume that γ > α for each
γ ∈ Xα by moving all the contradictory elements into X0, and the partition will still consist of I-positive sets. Next, define f : κ → κ by f(β) = α for which α is the unique ordinal such that β ∈ Xα. As f is a regressive function, weak normality kicks in to give γ∗ < κ such that
{α < κ : f(α) ≤ γ∗} ∈ I∗.
∗ ∗ To arrive at the contradiction, we fix η > γ and β ∈ Xη ∩ {α < κ : f(α) ≤ γ }. From the definition of f, we have
f(β) = η
as β ∈ Xη, yet
∗ β ∈ Xη ∩ {α < κ : f(α) ≤ γ } implies that f(β) ≤ γ∗. 45
This is the contradiction we are looking for, and so it follows that I is weakly κ- saturated. Conversely, suppose I is a semi-weakly normal, weakly κ-saturated ideal. Assume the contrary, and suppose I is not weakly normal. This means there is regressive function f : κ → κ with the property that for every γ < κ,
{α < κ : f(α) ≤ γ} 6∈ I∗.
Since f is a regressive function and I is a semi-weakly normal ideal, there is γ0 < κ such that
+ ∗ A0 = {α < κ : f(α) < γ0} ∈ I \ I .
∗ Notice, A0 cannot be an I because we are assuming I is not weakly normal. Since
+ κ \ A0 ∈ I , it follows that there is γ1 < κ such that
+ ∗ A1 = {α < κ : γ0 ≤ f(α) < γ1} ∈ I \ I .
We can continue to inductively build Aξ and γξ for ξ < κ. When ξ = ζ + 1, we know
+ κ \ Aζ ∈ I , so there is γζ+1 < κ such that
+ ∗ Aξ = {α < κ : γζ ≤ f(α) < γζ+1} ∈ I \ I .
When ξ < κ is a limit ordinal, we start with
ηξ = sup{γζ : ζ < ξ},
and using the fact that
+ {α < κ : ηξ ≤ f(α)} ∈ I ,
we can find γξ < κ such that
+ Aξ = {α < κ : ηξ ≤ f(α) < γξ} ∈ I .
The sequence hAξ : ξ < κi is a sequence of pairwise disjoint I-positive sets. To
get our contradiction, we need to show hAξ : ξ < κi is a partition of κ. This will be proved in the following claim. 46
CLAIM 3.3.1 The sequence hAξ : ξ < κi is a partition of κ into κ-many I-positive sets.
∗ Proof: Suppose β < κ, and our goal is to find ξ < κ such that β ∈ Aξ∗ . Let
∗ ξ = min{ξ : f(β) < γξ}.
We have to consider some cases based on ξ∗. If ξ∗ = 0, then
f(β) < γ0 which tells us that
β ∈ A0.
If ξ∗ = ζ + 1, then we have
γζ ≤ f(β) < γζ+1
which tells us that
β ∈ Aζ+1 = Aξ∗ .
If ξ∗ is a limit ordinal, then we have
ηξ ≤ f(β) < γξ∗ which gives us
β ∈ Aξ∗ .
Thus, we have that hAξ : ξ < κi is a partition of κ into κ-many I-positive sets, and the claim is proved. We were able to construct a partition of κ into κ-many I-positive sets which contradicts the fact that I is weakly κ-saturated. Therefore, I is weakly normal.
So, we have shown that an ideal will be weakly normal whenever it is semi-weakly normal and weakly κ-saturated. 47
A rather interesting corollary of this result is that every weakly normal ideal must be weakly κ-saturated, and this certainly false for normal ideals. For example, the non-stationary ideal, NS(κ), is a normal ideal that is not weakly κ-saturated. The non-stationary ideal cannot be weakly κ-saturated because every stationary subset can be partitioned into κ-many stationary subsets. If κ is a measurable cardinal, then κ has a κ-complete, normal ultrafilter, and an ultrafilter is trivially weakly κ- saturated. Now, we turn to our focus to least functions. Looking at the definition of a least function, condition (ii) is very similar to weak normality in that one function is dominating another function on a large set. With weak normality, regressive functions can be viewed as being dominated by the identity function. Also, the ideal J = f∗(I) is weakly normal for any ideal I with a least function f. These two observations provide motivation for considering least functions in more detail.
LEMMA 3.3.2 Suppose I is an ideal on a cardinal κ with a least function, f ∗. If I is weakly normal, then f ∗ is not regressive.
Proof: Suppose I is weakly normal, and we need to show that f ∗ is not regressive. If it was the situation that f ∗ was regressive, then from the weak normality of I, there is γ < κ such that {δ < κ : f ∗(δ) ≤ γ} ∈ I∗.
Yet, f ∗ is a least function, so it must happen that
{δ < κ : γ < f ∗(δ)} ∈ I+.
This means {δ < κ : f ∗(δ) ≤ γ} ∩ {δ < κ : γ < f ∗(δ)}= 6 ∅, and for any δ in the previous intersection, we have
f ∗(δ) ≤ γ < f ∗(δ), 48
a contradiction. Hence, f ∗ is not regressive.
We know that the identity function is a least function for a weakly normal ideal. The above result shows us that any least function for a weakly normal ideal must agree with the identity function on a set in I+. This is different than what happens in the case of normality. With normal filters, the behavior of a least function depends on other characteristics of the filter. For example, suppose F is a normal ultrafilter on a cardinal κ and fix a least function f. Then f cannot be regressive. The reason for this is if f was a regressive least function, then there is γ < κ such that
{δ < κ : f(δ) = γ} ∈ F.
Since F is an ultrafilter, we know {δ < κ : γ < f(δ)} ∈ F , and we have
{δ < κ : f(δ) = γ} ∩ {δ < κ : γ < f(δ)}= 6 ∅,
a contradiction. If {δ < κ : δ < f(δ)} ∈ F,
then as f is a least function, there is γ < κ such that
{δ < κ : δ ≤ γ} = [0, γ] ∈ F which implies F is improper. So, we must have that
{δ < κ : δ = f(δ)} ∈ F, and f agrees with the identity function almost everywhere. If a least function has a stronger failure of regression, then it turns out the ideal must be weakly normal. Next, we introduce a strong failure of regression and show the existence of a least function satisfying this property characterizes when an ideal is weakly normal. 49
DEFINITION 3.3.2 Let κ be a cardinal, and I is an ideal on κ. A function f is said to be non-regressive if {δ < κ : δ ≤ f(δ)} ∈ I∗.
Being a non-regressive function is a lot more difficult than being not regressive. If a function is not regressive, the places where the function fails to be regressive form an I-positive, where as a non-regressive function is not regressive almost everywhere. When an ideal has a non-regressive least function, the existence of this function forces the ideal to be weakly normal.
LEMMA 3.3.3 Suppose κ is cardinal and I is an ideal on κ. Then I has a non- regressive least function if and only if I is weakly normal.
Proof: Suppose f ∗ is a non-regressive least function for I. Let g : κ → κ be a regressive function. Our goal is to show that there is γ < κ such that
{δ < κ : g(δ) ≤ γ} ∈ I∗.
Since f ∗ is a non-regressive function and g is a regressive function, we know
δ ≤ f ∗(δ) and g(δ) < δ for each δ < κ. Putting these two facts together, we get
g(δ) < δ ≤ f ∗(δ) for each δ < κ. As f ∗ is a least function and
{δ < κ : g(δ) < f ∗(δ)} ∈ I∗, it follows that there is γ < κ such that
{δ < κ : g(δ) ≤ γ} ∈ I∗, 50 and this proves I is weakly normal. Conversely, suppose I is weakly normal. Since I is weakly normal, we know the identity function, d, is a least function. It is clear that d is non-regressive because
{δ < κ : δ ≤ d(δ)} = {δ < κ : δ ≤ δ} = κ ∈ I∗.
Hence, I has a non-regressive least function.
From Lemma 3.3.3, we get the following corollary relating weak normality and normality. In the corollary, we need to upgrade the least function to an exact upper bound for the constant functions.
DEFINITION 3.3.3 Let κ be a cardinal, and I is an ideal on κ. A function f ∗ : κ → κ is an exact upper bound (eub) for the constant functions modulo I if
i. For every γ < κ, {δ < κ : γ < f ∗(δ)} ∈ I∗.
ii. If g : κ → κ is a function such that {δ < κ : g(δ) < f ∗(δ)} ∈ I∗, then there is γ < κ such that {δ < κ : g(δ) ≤ γ} ∈ I∗.
Notice that every exact upper bound for the constant functions (modulo I) is a least function for I, but the converse is not true as it may happen that a least function is bounded on an I-positive set. Clearly, this cannot happen when dealing with exact upper bounds for the constant functions.
COROLLARY 3.3.1 If I is a normal ideal on κ and the constant functions have an exact upper bound modulo I, then I is weakly normal.
Proof: Let f ∗ : κ → κ be the exact upper bound for the constant functions. We will prove this corollary by showing that f ∗ is non-regressive, and then by Lemma 3.3.3, I will be weakly normal. To start, for each γ < κ, let
∗ Uγ = {δ < κ : γ < f (δ)}. 51
∗ ∗ Since f is an eub for the constant functions, we know Uγ ∈ I for each γ < κ. Now, the normality of I says
∗ F = 4γ<κUγ ∈ I .
For each δ ∈ F , we have δ ∈ Uγ for each γ < δ. The previous fact gives us
γ < f ∗(δ) for each γ < δ, and we have δ ≤ f ∗(δ).
Since δ ≤ f ∗(δ) is true for each δ ∈ F , we know
F ⊆ {δ < κ : δ ≤ f ∗(δ)}.
Hence, f ∗ is a non-regressive function, and I is weakly normal.
At the moment, it is unknown whether “f ∗ is an eub for the constant functions” can be replaced with “f ∗ is a least function.” In the proof, we used the fact that f ∗ dominates the constant functions, which may not be the case with a least function. In the presence of the exact upper bound for the constant functions, we have the following equivalencies with weak normality and the exact upper bound for the constant function being non-regressive.
LEMMA 3.3.4 Suppose κ is a cardinal, I is an ideal on κ, and f ∗ is an exact upper bound for the constant functions modulo I. Then the following are equivalent.
i. f ∗ is a non-regressive function.
ii. I is weakly normal.
iii. I is semi-weakly normal. 52
Proof: From our previous work, we know (i) ⇒ (ii) ⇒ (iii), so it remains to show that (iii) ⇒ (i). This almost happens immediately, for if f ∗ was not non-regressive, then f ∗ is regressive on an I-positive set. From the semi-weak normality of I, there is γ < κ such that {δ < κ : f ∗(δ) ≤ γ} ∈ I+.
Yet, the fact that f ∗ is an eub tells us that
{δ < κ : f ∗(δ) ≤ γ} ∈ I,
a contradiction. Hence, f ∗ must be a non-regressive function.
We turn now to the investigation of ideals with least functions that are not re- gressive, and show that they are “one set away” from being weakly normal.
LEMMA 3.3.5 Suppose I is a proper ideal on the cardinal κ. If I has a least function f that is not regressive, then there is a set A ⊆ κ so that the ideal J generated by I ∪ {κ \ A} is a weakly normal proper ideal.
Proof: Let A := {δ < κ : δ ≤ f(δ)},
all the places where f is not regressive. We will show the ideal, J = I ∪ {κ \ A}, is an ideal where f is non-regressive. To start with, we will show f is unbounded modulo
J. Let γ < κ be given and consider the set Uγ = {δ < κ : γ < f(δ)}. Assume the
∗ ∗ contrary and suppose that Uγ ∈ J. If Uγ ∈ J, then Bγ = κ \ Uγ ∈ J . Since Bγ ∈ J ,
we know A ∩ Bγ =I A. For any δ ∈ A ∩ Bγ, we have
δ ≤ f(δ) ≤ γ,
and since I is a uniform ideal, it follows that
|A ∩ Bγ| = κ. 53
+ These two facts contradict each other, so we have Uγ ∈ J . Next, suppose g : κ → κ is a function such that
{δ < κ : g(δ) < f(δ)} ∈ J ∗.
As {δ < κ : g(δ) < f(δ)} ∈ J ∗, we may assume that for all δ ∈ A that g(δ) < f(δ). Our goal is to find γ < κ such that
{δ < κ : g(δ) ≤ γ} ∈ J ∗.
To do this, we define the function h : κ → κ by g(δ), if δ ∈ A h(δ) = 0, if δ ∈ κ \ A
It’s clear that {δ < κ : h(δ) < f(δ)} ∈ I∗, and so there is γ < κ such that
{δ < κ : h(δ) ≤ γ} ∈ I∗.
Since I∗ ⊆ J ∗, we know {δ < κ : h(δ) ≤ γ} ∈ J ∗, which gives us A ∩ {δ < κ : h(δ) ≤ γ} ∈ J ∗.
For each δ ∈ A ∩ {δ < κ : h(δ) ≤ γ} ∈ J ∗, we have
g(δ) = h(δ) ≤ γ.
Hence, we have {δ < κ : g(δ) ≤ γ} ∈ J ∗, and f is a least function modulo J. The fact that f is a non-regressive function modulo J follows from the fact that A ∈ J ∗. Since J is an ideal with a non-regressive function modulo J, we have by Lemma 3.3.3 that J is weakly normal. 54
If it was the case that κ ∈ J, then as A ⊆ κ, it would follow that A ∈ J. This contradicts the fact that A ∈ J ∗. Thus, J is a proper ideal on κ.
In this section, we saw two different ways of creating a weakly normal idea. One method was using the image of a least function and the other method was to concen- trate the ideal around the set of ordinals where the least function was not regressive. For a least function f, let
N = {δ < κ : δ ≤ f(δ)}.
There are three possibilities for N: N ∈ I, N ∈ I∗, or N ∈ I+ \ I∗. No matter how f behaves, we know f∗(I) will be weakly normal. Also, f∗(I) will be proper because
f −1(κ) = κ 6∈ I.
But, the ideal I ∪ {κ \ A} will change depending on N.
• If N ∈ I or f is regressive, then I ∪ {κ \ N} = P (κ). The ideal grows so much that it becomes improper.
• If N ∈ I∗ or f is non-regressive, then I ∪ {κ \ N} = I. In this case, the ideal does not grow at all.
• If N ∈ I+ \ I∗, then I ∪ {κ \ N} is an ideal that is strictly larger than I but still proper.
These observations about f∗(I) and I ∪ {κ \ N} lead to the following question.
QUESTION 3.3.1 Suppose I is an ideal on κ, and f is a least function modulo I.
What relation is there between the ideals f∗(I) and I ∪ {κ \ N}? 55
4 Partition Results
4.1 Motivation For The Main Theorem
In this chapter, we prove our main theorem which gives a coloring theorem for inaccessible cardinals possessing a stationary subset which does not reflect in inaccessibles. Our motivation is the following result of Shelah (Claim 3.3 in [13]):
THEOREM 4.1.1 (Shelah, [13]) Suppose κ is an inaccessible cardinal and there
2 is a stationary subset of κ that does not reflect in the inaccessibles, then κ 6→ [κ]σ for every σ < κ.
Given the above, it is natural to ask if we can upgrade this result to obtain
2 κ 6→ [κ]κ? The main theorem gives an “approximate” yes to this question. It is possible to construct a coloring c :[κ]2 → κ with the property that the range of every unbounded subset of κ is almost all of κ relative to a certain naturally defined
2 club-guessing ideal. This ideal must be weakly κ-saturated if κ → [κ]κ. The main theorem and its corollary are analogues of results obtained by Eisworth in [4] for successors of singular cardinals. We also explore how much of Eisworth’s work in [4] can be implemented in this context. Our coloring will be constructed using Todorˇcevi´c’smethod of minimal walks, and we spend the first part of the chapter discussing this technique. After that, we begin to integrate the club-guessing result from Section 2.2 with ideas pioneered by Eisworth and Shelah [4, 7] before moving on to the proof of our main theorem.
4.2 Minimal Walks
The method of minimal walks was created and used by Todorˇcevi´cin [15]. Todorˇcevi´cused minimal walks to create Aronszajn trees and prove ℵ 6→ [ℵ ]2 . 1 1 ℵ1 56
2 Actually, Todorˇcevi´c’sproof showed λ 6→ [λ]λ whenever λ is a regular uncountable cardinal with a non-reflecting stationary subset as noted by Shelah in [12]. For a
λ+ + regular cardinal λ, the set Sλ is a non-reflecting stationary subset of λ , and so
+ + 2 λ 6→ [λ ]λ+ . Our notation for minimal walks will be hybrid of Todorˇcevi´c’sand Shelah’s nota- tion.
DEFINITION 4.2.1 Fix a regular cardinal κ > ω. Let e = heα : α < κi be a
sequence such that each eα is club in α. Suppose α < β < κ. Define the natural
number ρ2(α, β) and sequence of ordinals hβi(α, β): i ≤ ρ2(α, β)i as follows.
For l = 0, we set β0(α, β) = β
If βn(α, β) = α, then we set ρ2(α, β) = n, otherwise
If βl(α, β) > α, then we define
βl+1(α, β) = min(eβl(α,β) \ α).
A minimal walk is a non-increasing sequence of ordinals, so it must be eventually
constant. The sequence hβl(α, β): l ≤ ρ2(α, β)i is called the upper walk or the walk from β to α through e. As we walk from β to α through e, we define another sequence called the lower walk.
DEFINITION 4.2.2 Fix a regular cardinal κ > ω. Let e = heα : α < κi be a
sequence such that each eα is club in α. Suppose α < β < κ. Let hβl(α, β): l ≤
− ρ2(α, β)i be the walk from β to α. We define the ordinals βl (α, β) for l < ρ2(α, β) as follows.
− For l = 0, let β0 (α, β) = 0, and 57
For l + 1 ≤ ρ2(α, β),
− βl+1(α, β) = sup(eβl(α,β) ∩ α).
For each l + 1 < ρ2(α, β), the ordinal α is not a member of eβl(α,β), so βl+1(α, β)
− and βl+1(α, β) are two ordinals in eβl(α,β) that sandwich α, and so
− (βl+1(α, β), βl+1(α, β)) ∩ eβl(α,β) = ∅.
Even though the upper walk is a non-increasing sequence of ordinals, the lower walk is less predictable but still important in our proofs. To gain some control over the lower walk, we define for k ≤ ρ2(α, β),
− γk(α, β) = max{βl (α, β): l ≤ k}.
th The ordinal γk(α, β) tells us the largest step of the lower walk up to the k step.
− As βl (α, β) < α for all l < ρ2(α, β), we have for k < ρ2(α, β),
− γk(α, β) = max{βl (α, β): l ≤ k} < α and
γρ (α,β)(α, β) < α ⇐⇒ α ∈ nacc(eβ (α,β)). 2 ρ2(α,β)
One major and useful property about minimal walks is the following proposition.
∗ PROPOSITION 4.2.1 For k ≤ ρ2(α, β), if γk(α, β) < α ≤ α, then
∗ hβl(α, β): l ≤ ki = hβl(α , β): l ≤ ki.
∗ Proof: For k = 0, we have β0(α, β) = β = β0(α , β). Suppose for l < k that
∗ βl(α, β) = βl(α , β).
∗ ∗ Now, we show for l + 1 ≤ k that βl+1(α, β) = βl+1(α , β). Since γk(α, β) < α ≤ α, we know
∗ sup(eβl(α,β) ∩ α) < α ≤ α, 58 so
∗ min(eβl(α,β) \ α) = min(eβl(α,β) \ α ).
Using these facts, we have
βl+1(α, β) = min(eβl(α,β) \ α)
∗ = min(eβl(α,β) \ α )
∗ ∗ = min(eβl(α ,β) \ α )
∗ = βl+1(α , β) and our first part is proved.
From the previous proposition, we have the following corollary.
∗ COROLLARY 4.2.1 If γρ2(α,β)(α, β) < α, then for any α satisfying
∗ γρ2(α,β)(α, β) < α ≤ α, we have
∗ hβl(α, β): l ≤ ρ2(α, β)i = hβl(α , β): l ≤ ρ2(α, β)i.
Moreover,
∗ βρ2(α,β)(α , β) = α.
Proposition 4.2.1 tells us IF it happens that γρ2(α,β)(α, β) < α, THEN the first
∗ ρ2(α, β) steps of the walk from β to α is identical to the first ρ2(α, β) steps of the
th ∗ walk from β to α AND the ρ2(α, β) step of the walk from β to α is α. In other words, the walk from β to α∗ extends the walk from β to α. Using Proposition 4.2.1, we get the following corollary.
∗ COROLLARY 4.2.2 For k ≤ ρ2(α, β), if γk(α, β) < α ≤ α, then
− − ∗ hβl (α, β): l ≤ ki = hβl (α , β): l ≤ ki. 59
The combination of Proposition 4.2.1 and Corollary 4.2.2 give a “safe interval” of ordinals that extend a minimal walk. When proving the constructed coloring works, this “safe interval” will be crucial in controlling the coloring.
4.3 The Ideal idp(C, I)
In this section, we lay the groundwork for the main theorem. First, we introduce
the reader to the ideal idp(C, I), originally defined by Shelah in [13]. The ideal
idp(C, I), a creation of Shelah, has played a major role in coloring theorems in the other situations. The coloring we construct will have the property that any unbounded
subset of κ will have a large range when measured by idp(C, I). After the introduction
of our special ideal and characterizing the large sets of idp(C, I), we construct a special C-sequence which we will use for our minimal walks.
κ Let κ be an inaccessible cardinal, ω < cf(λ) = λ < κ, and S ⊆ Sλ be a stationary subset that does not reflect in any inaccessible. Suppose C = hCδ : δ ∈ Si is an S-club system such that
• Cδ has order-type λ, and
• For any club E ⊆ κ, there exists stationary δ ∈ S such that
sup{cf(): ∈ E ∩ Cδ} = δ.
The existence of such S-club guessing sequence is guaranteed from Lemma 2.2.1
and Lemma 2.2.2. For each δ ∈ S, let Iδ be an ideal on Cδ defined by
Iδ = {B ⊆ Cδ : sup{cf(): ∈ B} < δ}.
Let I be the ideal idp(C, I), defined by putting A ∈ I if there is a club E ⊆ κ such that
δ ∈ S ∩ E ⇒ A ∩ E ∩ Cδ ∈ Iδ. 60
Before we get to defining a coloring and showing the range of our coloring is a large set (relative to I), we will look at what it means to be almost everything with respect to I.
OBSERVATION 4.3.1 Let A ⊆ κ. A ∈ I∗ if and only if there is a club E ⊆ κ such that for every δ ∈ S ∩ E, there is γ∗ < δ such that
∗ E ∩ nacc(Cδ) \ γ + 1 ⊆ A.
Proof: If A ∈ I∗, then there is a club E ⊆ κ such that for each δ ∈ S ∩ E we have
(κ \ A) ∩ E ∩ Cδ ∈ Iδ.
The club E we get from this fact will be the required club, so let δ ∈ S ∩ E. Since we know
(κ \ A) ∩ E ∩ Cδ ∈ Iδ,
it follows that (κ \ A) ∩ E ∩ Cδ is contained in a finite union of elements that generate
∗ Iδ, call it U. So, let γ be the bound on the set of non-accumulation points of Cδ in U. Next, if
∗ α ∈ E ∩ nacc(Cδ) \ γ + 1,
then α ∈ A, for if this is not the case, then we would have a non-accumulation point of
Cδ that is an element of (κ\A)∩E greater than the bound on such non-accumulation points, a contradiction. Conversely, suppose there is a club E ⊆ κ such that for each δ ∈ S ∩ E there is a γ∗ < δ such that
∗ E ∩ nacc(Cδ) \ γ + 1 ⊆ A.
We will show that the same E witnesses A ∈ I∗, so let δ ∈ E∩S. From our hypothesis,
∗ ∗ there is γ < δ such that E ∩ nacc(Cδ) \ γ + 1 ⊂ A. If α ∈ (κ \ A) ∩ E ∩ Cδ, then it follows that
∗ α 6∈ E ∩ nacc(Cδ) \ γ + 1. 61
∗ This occurs either if α ∈ acc(Cδ) or α ≤ γ as α ∈ E ∩ Cδ. This means
∗ (κ \ A) ∩ E ∩ Cδ ⊆ acc(Cδ) ∪ (γ + 1),
∗ which gives tells us that (κ \ A) ∩ E ∩ Cδ ∈ Iδ. Hence, we have A ∈ I .
With Observation 4.3.1 out of the way, we focus our attention on constructing a C-sequence that will be used in our minimal walks. We use a technique originating with Shelah where our C-sequence is made to “swallow” our S-club guessing sequence.
What this means is for our C-sequence, heα : α < κi, if δ ∈ S ∩ eα, then
Cδ \ eα is bounded in δ.
This swallowing allows us to use the S-club guessing sequence in our minimal walk and control our coloring.
PROPOSITION 4.3.1 Suppose κ is an inaccessible cardinal, ω < cf(λ) = λ < κ,
κ S ⊆ Sλ is a stationary subset that does not reflect in any inaccessible, and C = hCδ : δ ∈ Si is an S-club system such that otp(Cδ) = λ. Then there is a C-sequence heα : α < κi such that
i). If α is a regular cardinal, then eα ∩ S = ∅,
ii). If cf(α) < α, then |eα| < min(eα) and |eα| ≤ λ + cf(α), and
iii). If δ ∈ S ∩ eα, then Cδ \ eα is bounded in δ.
Proof: Without loss of generality, we may assume S contains only cardinals. To start
∗ with, we define heα : α < κi as follows:
∗ ∗ (a). If α is an inaccessible cardinal, then let eα be club in α with S ∩ eα = ∅,
(b). If α is a cardinal that is the successor of a cardinal ξ, then eα ⊂ (ξ, α) is club in α, 62
(c). If α is the successor of the ordinal β, then eα = {β},
∗ (d). If α is a limit ordinal that is not an inaccessible cardinal, let eα be a club in α
∗ such that otp(eα) = cf(α).
If X ⊆ α for some α, we denote the closure of X in α by cl[X]α. For each k < ω, we define heα[k]: α < κi in the following manner:
∗ eα[0] = eα, and
S eα[k + 1] = cl[eα[k] ∪ {Cδ : δ ∈ S ∩ eα[k]}]α.
0 S Now, for each α < κ, we define eα = cl[ k<ω eα[k]]α, and 0 |eα| u 1, if cf(α) < α, α∗ = 0, otherwise .
0 ∗ Finally, we set eα = eα \ α .
It remains to show that heα : α < κi has the desired properties. First, suppose α
∗ ∗ is an regular cardinal. From the way eα was picked, we know eα ∩ S = ∅, and from that we have
S eα[k + 1] = cl[eα[k] ∪ {Cδ : δ ∈ S ∩ eα[k]}]α
= cl[eα[k] ∪ ∅]α
= cl[eα[k]]α
= eα[k].
Using this fact with the definition of eα, we have eα ∩ S = ∅. The second fact follows from the definition of α∗ when α is singular.
0 ∗ For the third requirement, suppose η ∈ eα ∩S. As η ∈ eα = eα \α and cf(η) = λ, there is k < ω such that η ∈ eα[k]. This means η ∈ eα[k] ∩ S, and this in turn implies
S 0 ∗ Cη ⊆ {Cδ : δ ∈ S ∩ eα[k]} ⊆ eα. Since η is a limit ordinal and η > α , we have
∗ 0 ∗ Cη \ α ⊂ eα \ α = eα. Hence, we have that Cη \ eα is bounded in η. 63
Condition (iii) is critical in the proof of our partition theorem. The trick of having a C-sequence “swallow” a given club-guessing sequence was first utilized by Shelah in Chapter III of [13]. This “swallowing” trick allows us to import, in a way, characteristics of a club-guessing sequence into a C-sequence. The importing of the club-guessing characteristic allow us to pick a specific ordinal in the “safe interval”.
4.4 The Main Theorem
Now, we will construct a coloring for which the image of any unbounded subset of
κ is a large set with respect to idp(C, I). With the C-sequence from Section 3, many of the ideas from Eisworth [4] can be carried over to our situation.
THEOREM 4.4.1 (Main Theorem) Suppose
i). κ is an inaccessible cardinal,
ii). ω < λ = cf(λ) < κ,
κ iii). S ⊆ Sλ is a stationary subset that does not reflect in any inaccessible, and
iv). C = hCδ : δ ∈ Si is a club system such that otp(Cδ) = λ and for every club E ⊆ κ there exists stationarily many δ ∈ S such that for each α < δ there is
∈ E ∩ Cδ with cf() > α.
Then there is c :[κ]2 → κ such that for every unbounded subset A ⊆ κ, we have
2 ∗ rng(c [A] ) ∈ I where I = idp(C, I) and I = hIδ : δ ∈ Si with Iδ = {B ⊆ Cδ : sup{cf(): ∈ B} < δ}.
Proof: Let heα : α < κi be the C-sequence from Proposition 4.3.1 using the provided
2 hCδ : δ ∈ Si. Now, we define c :[κ] → κ as follows: 64
For each α < β < κ, let k(α, β) ≤ ρ2(α, β) be maximal for which
− − hβl (α, β): l ≤ k(α, β)i = hβl (γk(α, β) + 1, α): l ≤ k(α, β)i.
Define c(β, α) = βk(α,β)(β, α).
Our next goal is to show the coloring will do its job. Let A ∈ [κ]κ. From Observation
2 ∗ 4.3.1, we know rng(c [A] ) ∈ I if and only if there is a club E ⊆ κ such that for all δ ∈ S ∩ E there is γ∗ < δ such that
∗ 2 nacc(Cδ) ∩ E \ γ + 1 ⊆ rng(c [A] ).
To define such club, we fix a κ-approximation sequence hMi : i < κi over {κ, A, C, e}.
This means hMi : i < κi is a continuous ∈-chain of elementary submodels of H(χ), for a sufficiently large regular cardinal χ, such that
• κ, A, C and e are all in M0,
• |Mi| < κ,
• hMj : j ≤ ii ∈ Mi+1 for all i < κ, and
• Mi ∩ κ is a proper initial segment of κ for each i. Using this sequence, we define
E = {δ < κ : Mδ ∩ κ = δ},
which is club in κ. The overall goal is to show the club E satisfies Observation 4.3.1.
So, let δ ∈ S ∩ E with nacc(Cδ) ∩ E unbounded in δ. Note, if nacc(Cδ) ∩ E was
∗ ∗ bounded in δ, then let γ be the bound, and trivially, we have nacc(Cδ) ∩ E \ γ + 1 ⊆
2 rng(c [A] ).
Pick β ∈ A such that β > Mδ+1 ∩ κ, and then we walk down from β to δ through e. Let
∗ β = βρ2(δ,β)−1(δ, β), 65
the second to last step of the walk down from β to δ. Since δ ∈ eβ∗ ∩ S, we know
Cδ \ eβ∗ is bounded in δ (this follows from property (iii) of e) and denote this bound
∗ by νδ. We haven’t found our γ , but we will soon. Let
∗ η = max{νδ, γρ2(δ,β)−1(δ, β)} + 1,
and set
∗ ∗ γ = min(Cδ \ η ).
∗ Suppose ∈ nacc(Cδ) ∩ E \ γ + 1. Now, our immediate task is to find α ∈ A such that c(α, β) = . Before we do this, there are a couple of things we want to take stock of.
One fact is ∈ nacc(eβ∗ ). Since ∈ nacc(Cδ) and > νδ, we know
∈ (Cδ \ νδ) = (eβ∗ \ νδ),
and ∈ eβ∗ . Since ∈ nacc(Cδ), we also know sup(Cδ ∩ ) < . If ∈ acc(eβ∗ ),
then there would be ν ∈ eβ∗ such that sup(Cδ ∩ ) < ν < . But such ν would be an
element of Cδ greater than sup(Cδ ∩ ), a contradiction. Hence, ∈ nacc(eβ∗ ). Now, since
γρ2(δ,β)−1(δ, β) < < δ,
we have
hβl(δ, β): l < ρ2(δ, β)i = hβl(, β): l < ρ2(δ, β)i,
which follows from Proposition 4.2.1. From this, it follows that
∗ β = βρ2(δ,β)−1(δ, β) = βρ2(δ,β)−1(, β).
From the fact that ∈ eβ∗ , we have
βρ (δ,β)(, β) = min(eβ (δ,β) \ ) 2 ρ2(δ,β)−1
= min(eβ∗ \ ) = . 66
th So, the ρ2(δ, β) step on the walk down from β to is , which means
ρ2(δ, β) = ρ2(, β).
From the fact that ∈ nacc(eβ∗ ), we glean two more facts. First, we have
− β (, β) = sup(eβ ∩ ) = sup(eβ∗ ∩ ) < . ρ2(,β) ρ2(,β)−1
Second, we have γ (, β) = β− (, β). ρ2(,β) ρ2(,β)
∗ The second fact is true because γ ∈ Cδ \ νδ and Cδ \ νδ = (eβ∗ \ νδ) ∩ δ, and we get
γ (, β) < γ∗ ≤ β− (, β) ≤ γ (, β). ρ2(,β)−1 ρ2(,β) ρ2(,β)
Now, define − := β− (, β) = γ (, β). ρ2(,β) ρ2(,β)
Notice, whenever − < α ≤ , we have
hβl(α, β): l ≤ ρ2(, β)i = hβl(, β): l ≤ ρ2(, β)i
− − hβl (α, β): l ≤ ρ2(, β)i = hβl (, β): l ≤ ρ2(, β)i, and
βρ2(,β)(α, β) = .
We are almost done with our proof. So far, we have shown for any ordinal between − and , the walk down from β to α will extend the walk from β to and must pass through . Our next step is to carefully pick α ∈ A between − and to ensure c(α, β) = . To do this, we will make use of our club E. First, let
k = ρ2(, β) 67
and
− s = hβl (, β): l ≤ ki.
Let ϕ(x, y) be the formula which states • x < y • y ∈ A
• ρ2(x, y) = k
− • hβl (x, y): l ≤ ki = s From our work so far, we know that ϕ(, β) holds. Also, all the parameters needed to define ϕ lie in the model M. Here is where we make use of our club E. Since and δ are both in E, we have
= M ∩ κ < δ = Mδ ∩ κ < β.
Now, we will show (∃statx < κ)(∃∗y < κ)[ϕ(x, y)].
In this context, ∃∗y < κ means “there exists unboundedly many y < κ such that ϕ(x, y)” and ∃statx < κ means “there are stationarily many x < κ such that (∃∗y < κ)[ϕ(x, y)].” Let X = {x < κ :(∃∗y < κ)[ϕ(x, y)]}. We want to show that for any club F ⊆ κ, we have X ∩ F 6= ∅. Since F is club in κ and = M ∩ κ, we know F ∩ is club in
which implies that ∈ F . Since ϕ(, β), < β < κ and < δ = Mδ ∩ κ, it follows that
Mδ |= (∃y < κ)( < y ∧ ϕ(, y)).
From the above comment, there is y ∈ A such that
< y < δ and ϕ(, y). 68
Since y < δ < β, we know that
∗ Mδ |= (∃ z < κ)( < z ∧ ϕ(, z)).
As Mδ is a elementary submodel of H(χ), it follows that
(∃∗y < κ)(ϕ(, y)).
Hence, ∈ {x < κ :(∃∗y < κ)[ϕ(x, y)]} and X is stationary. Thus,
(∃statx < κ)(∃∗y < κ)[ϕ(x, y)].
stat ∗ ∗ Since (∃ x < κ)(∃ y < κ)[ϕ(x, y)] holds in M, we can find and α such that
− < ∗ < α < ,
ϕ(∗, α)
and
∗ e ∩ ( , α) 6= ∅.
We finally can prove c(α, β) = . This portion of the proof is just as in Eisworth [4].
CLAIM 4.4.1 c(α, β) = .
− ∗ Proof: As < < α < , we know the first ρ2(, β) steps of the upper and lower walk from β to α agree with the upper and lower walk from β to . We get c(α, β) =
if we can show that k(α, β) = ρ2(, β). This will be completed in a two claims.
CLAIM 4.4.2 If k ≤ ρ2(, β), then
∗ (1) γk( , α) = γk(α, β), and
− − (2) hβl (α, β): l ≤ ki = hβl (γk(α, β) + 1, α): l ≤ ki. 69
∗ Proof: As k ≤ ρ2(, β) and ϕ( , α) holds, we know
∗ γk( , α) = γk(, β) = γk(α, β).
The second equality holds because the first k steps of the lower walk from β to α are the same as the first k steps of the lower walk from β to . For the second part of the claim, we have
− − hβl (α, β): l ≤ ki = hβl (, β): l ≤ ki by Corollary 4.2.2
− ∗ ∗ = hβl ( , α): l ≤ ki by ϕ( , α)
− ∗ = hβl (γk( , α) + 1, α): l ≤ ki by Corollary 4.2.2 − = hβl (γk(α, β) + 1, α): l ≤ ki by Claim 4.4.2 (1)
This finishes the claim, and it is time for the second claim.
CLAIM 4.4.3 If k = ρ2(, β) + 1, then
− − hβl (α, β): l ≤ ki= 6 hβl (γk(α, β) + 1, α): l ≤ ki.
Proof: For this k, we have
− β (α, β) = sup(eβ ∩ α) = sup(e ∩ α), k ρ2(,β)
∗ ∗ and we picked and α so that e ∩ ( , α) 6= ∅. With this careful choice, we know that
− − < sup(e ∩ α) = βk (α, β), and this choice of α gives
∗ − < sup(e ∩ α) = βk (α, β) = γk(α, β).
By way of contradiction, suppose that
− − hβl (α, β): l ≤ ki = hβl (γk(α, β) + 1, α): l ≤ ki. 70
If these two sequences are equal, then
− − hβl (γk(α, β) + 1, α): l ≤ ρ2(, β)i = hβl (α, β): l ≤ ρ2(, β)i assumption − = hβl (, β): l ≤ ρ2(, β)i Corollary 4.2.2
− ∗ ∗ = hβl ( , α): l ≤ ρ2(, β)i ϕ( , α) holds. We have
∗ hβl(γk(α, β) + 1, α): l ≤ ρ2(, β)i = hβl( , α): l ≤ ρ2(, β)i.
If these two sequences are equal, then we get
∗ ∗ βk−1(γk(α, β) + 1, α) = βk−1( , α) = ,
yet we have
∗ βk(γk(α, β) + 1, α) ≤ βk−1(γk(α, β) + 1, α) = .
∗ But, this contradicts the fact that γk(α, β) > .
Combining the previous two claims, we have c(α, β) = . Hence, we have shown
∗ 2 nacc(Cδ) ∩ E \ γ + 1 ⊆ rng(c [A] ). Thus, we have shown that for any unbounded 2 subset A of κ, we have rng(c [A] ) is an element of the dual filter.
The last theorem showed that the coloring changes any unbounded subset of κ into a large set relative to the ideal idp(C, I). While this is a nice piece of information about our ideal, we get more out of our ideal by adding one more hypothesis. The following corollary ties together a square-bracket relation and weak saturation.
COROLLARY 4.4.1 Let C and I be as in Theorem 4.4.1. Suppose κ is an inac-
κ cessible cardinal, and κ has a stationary subset S ⊆ Sλ with ω < cf(λ) = λ < κ that
2 does not reflect in any inaccessible. If κ → [κ]κ, then there is an ideal I such that • I is a uniform ideal on κ, extending the non-stationary ideal, • I is σ-indecomposable for all sufficiently large regular σ < κ, and • I is weakly κ-saturated. 71
Proof: Let I be the ideal idp(C, I). From Observation 3.2 (1) and (2) in [13], we know I meets the first two requirements. It remains to show that I is weakly κ-saturated. Suppose I is not weakly κ-saturated, and let p : κ → κ be a partition of κ such that p−1({α}) ∈ I+ for each α < κ. Let c :[κ]2 → κ be the coloring from Theorem 4.4.1, and we will
2 show the function p ◦ c will give us κ 6→ [κ]κ. Let A ∈ [κ]κ and ξ < κ be given. Our goal is to find α, β ∈ A such that
2 ∗ p ◦ c({α, β}) = ξ. From Theorem 4.4.1, we know rng(c [A] ) ∈ I , so we can pick α, β ∈ A such that c(α, β) ∈ p−1({ξ}).
Then, for such α, β, we have
p ◦ c({α, β}) = p(c({α, β})) = ξ.
2 Hence, κ 6→ [κ]κ, the contradiction we are looking for. Thus, I is weakly κ-saturated. 72
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