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Club-guessing, stationary reflection, and coloring theorems

We obtain strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. Along the way, we establish new results in club-guessing and in the general theory of ideals.


Introduction
Our main results establish that certain failures of simultaneous reflection for stationary subsets of the successor of a singular cardinal can be used to generate strong versions of negative square-brackets partition relations. This represents a substantial improvement over previous results -we get stronger conclusions from much weaker hypotheses. We are able to obtain this by way of progress in two areas. First, our primary advance allows us to obtain a version of the main result of [7] which holds even for successors of singular cardinals of countable cofinality. We accomplish this by changing the sort of club-guessing sequence used in those papers, and by refining our arguments so that they work in this new context. On another front, we extend results of Shelah connecting the structure of ideals with simultaneous reflection of stationary sets, and then combine this with the other advance to deduce the main theorem.
In this introductory section, our goal is to provide enough background so that we are both able to state our main result precisely and situate it in context. We assume the reader has an acquaintance with basic set-theoretic notation; any of the standard references (say [13], [19], or [15]) are more than sufficient. Some of the objects of importance to us have been denoted in a variety of ways throughout the literature, so we take a moment to set our conventions.
One more convention is quite important for us: by "I is an ideal on κ", we mean "I is a proper ideal on κ containing all bounded subsets of κ". We have chosen this notation for convenience, since it eliminates many trivialities.
We now move on to the background material; we begin with the square-brackets partition relation κ → [λ] µ θ of Erdős, Hajnal, and Rado [10], which means that for any function F : [κ] µ → θ, (which we refer to as a coloring) we can find a set H ⊆ κ of cardinality λ for which ran(F ↾ [H] µ ) θ, that is, when we restrict the function F to [H] µ , at least one color is omitted. These square-brackets partition relations arise naturally when one investigates the extent to which Ramsey's Theorem generalizes to uncountable sets, as the statement "κ → [λ] µ θ " asserts that a very weak form of Ramsey's Theorem holds at the cardinal κ.
The negations of these partition relations are quite strong combinatorial hypotheses in their own right, and it is for this reason that they have been studied extensively by set theorists: The moral is that if one knows an ordinary negative partition relation then it is often worth asking whether a stronger assertion, a negative square bracket relation, is also true. [11] Consider for example, the statement κ [κ] 2 κ . This asserts the existence of a function F : [κ] 2 → κ (a coloring of the pairs from κ using κ colors) with the property that F assumes every possible value on (the pairs from) every unbounded subset of κ. This says that Ramsey's Theorem fails in a very spectacular way at κ: it is possible to color the pairs of ordinals less than κ utilizing κ colors such that any subset of κ of size κ is completely inhomogeneous with respect to the coloring. Theorems that assert the existence of "complicated" colorings of pairs (or other finite sets) of ordinals (where the exact meaning of "complicated" depends on context) can be conveniently grouped under the sobriquet "coloring theorems", and our main theorem is one such.
In Appendix 1 of [26], Shelah systematically studies several combinatorial principles stronger than the negated square-brackets relations discussed above, and our results are most naturally stated using his notation. Definition 1.1. If λ is an infinite cardinal, and κ + θ ≤ µ ≤ λ, then Pr 1 (λ, µ, κ, θ) asserts the existence of function c : [λ] 2 → κ such that whenever we are given a collection t α : α < µ of pairwise disjoint elements of [λ] <θ as well as an ordinal ς < κ, then there are α < β for which c ↾ t α × t β is constant with value ς.
Notice that for θ > 2, the relation Pr 1 (λ, µ, κ, θ) implies λ [µ] 2 κ , as one may take the sets t α to be singletons; this justifies our referring to theorems that establish instances of Pr 1 as coloring theorems.
We will break off our discussion of these relations for a moment to pick up another thread that is important for our work, namely, reflection of stationary sets. Jech's [12] gives a nice introduction to this topic, and the paper [5] of Cummings, Foreman, and Magidor is also an excellent resource. Definition 1.2. Let S be a stationary subset of an uncountable regular cardinal κ.
(1) We say S reflects at α if α < κ has uncountable cofinality and S ∩ α is stationary in α.
(2) Refl(S) holds if every stationary subset of S reflects at some α.
(3) S is non-reflecting if S does not reflect at any α.
We observe here that any cardinal of the form κ + for regular κ has a nonreflecting stationary subset: consider the set S κ + κ of all ordinals below κ + of cofinality κ. This set does not reflect, as any α < κ of uncountable cofinality contains a closed unbounded subset of ordinals each of cofinality less than κ. The situation at successors of singular cardinals is much more delicate, and we will have much to say about this later.
It has been known for a long time that there are connections between stationary reflection and coloring theorems. For example, Tryba [30] (and independently Hugh Woodin) established that κ [κ] <ω κ whenever κ has a non-reflecting stationary subset. Stevo Todorčević [28] was able to use his technique of minimal walks to improve this to colorings of pairs: κ [κ] 2 κ whenever κ has a non-reflecting stationary set, and hence, in particular, κ + [κ + ] 2 κ + whenever κ is a regular cardinal. Shelah [25] simplified Todorčević's argument a bit and obtained some other generalizations.
Research on coloring theorems for successors of regular cardinals has continued. For example, Shelah [27] has established that Pr 1 (κ +2 , κ +2 , κ +2 , κ) holds for every regular cardinal κ, while Justin Moore [22] obtained a significant strengthening of Todorčević's result for ℵ 1 and solved a long-standing open problem in general topology.
Given our understanding of successors of regular cardinals, the following question is natural: To what extent are analogous results true for successors of singular cardinals?
This question is still very mysterious, in large part due to the fact that the combinatorics of successors of singular cardinals is very sensitive to underlying assumptions in set theory. For example, if µ is singular and µ + has a non-reflecting stationary subset, then the work of Todorčević can be brought to bear and we can conclude µ + [µ + ] 2 µ + . However, assuming the existence of large cardinals, Magidor [21] established the consistency of Refl(µ + ) for µ, and so the situation at successors of singular cardinals differs greatly from that at successors of regular cardinals.
Notice that this is a strong negative partition relation using cf(µ) colors; the question of whether or not this can be improved to a coloring using µ + colors (that is, whether Pr 1 (µ + , µ + , µ + , cf(µ)) holds for µ singular) is still very much open, as are the related questions concerning µ + [µ + ] 2 µ + and µ + [µ + ] <ω µ + . The main result of this paper obtains strong colorings under very weak assumptions, but to describe the precise situation, we need some notation from [5]. Definition 1.3. Let κ be a regular uncountable cardinal, and let S be a stationary subset of κ. Then Refl(θ, S) holds if for every sequence T i : i < θ of stationary subsets of S, there exists an α < κ such that each T i reflects at α. Similarly, we say Refl(< θ, S) holds if for every σ < θ and every sequence T i : i < σ of stationary subsets of S, there exists an α < κ such that each T i reflects at α.
We can now state our main theorem, a theorem illustrating the connection between failures of simultaneous reflection of stationary sets and the existence of complicated colorings.
Main Theorem. Assume µ is a singular cardinal.
(4) If cf(µ) = ℵ 0 and Refl(< cf(µ), S µ + ≥θ ) fails for arbitrarily large θ < µ, then there is a function d : [µ + ] 3 → µ + such that whenever t α : α < µ + is a pairwise disjoint family of finite subsets of µ + and ς < µ + , there are α < β < γ such that The proof builds on work of the author and Shelah [8,9,7]. In particular, the main theorem of [7], when taken together with results in sections 2 and 3 of the current paper, is strong enough to imply our main theorem in the case that µ has uncountable cofinality. The paper [9] partially extended the results of [7] to the case where µ has countable cofinality (the combination of the main theorem in [9] with the results of sections 2 and 3 in the current paper results in something weaker than Pr 1 (µ + , µ + , µ + , cf(µ))); the current paper gets around the obstacle in [9] caused by the countable cofinality of µ.
The proof of most parts of the main theorem will proceed according to the following sketch:
(1) In ZFC we prove that there are a function c : [λ] 2 → λ and an ideal I (related to club-guessing) such that whenever t α : α < λ is a family of pairwise disjoint elements of [λ] <cf(µ) , for almost every (in the sense of I) β * < λ there are α < β such that c ↾ t α × t β is constant with value β * .
(2) If the desired colorings fail to exist, then we can conclude ideal I from (1) possesses some strong combinatorial properties.
(3) These properties are strong enough to imply the needed instances of simultaneous reflection of stationary subsets of λ.
The main theorem of [7] establishes (1) for the case where the cofinality of µ is uncountable, while the main theorem of [9] gives a conclusion weaker than (1). Statement (2) will be obtained from (1) by modifying one of the arguments from [7]. Statement (3) relies on some extending some results from section 3 of Chapter IV of Shelah's [26]; the results seem to be new, though they have a "folklore-ish" flavor.
Moving on, we now give an account of the organization of this paper. In sections 2 and 3, we prove a series of results culminating in a proof of (3) above. Section 4 proves a club-guessing result needed to define the ideal I. Section 5 pins down the assumptions we need for our main theorems, and lays some groundwork for later arguments as well. Section 6 introduces much of the minimal walks machinery we need, as well as providing a proof of a crucial preliminary result. Section 7 furnishes some background material concerning scales and elementary submodels, and Section 8 provides a proof of (1), as well as a few other odds and ends we use in the proof of our main theorem. Section 9 closes the paper by finishing the proof of our primary result.
Finally, some historical remarks are in order. Clearly, the research in this paper rests on that of Saharon Shelah, and although the main theorem is an advance in our knowledge, the proof is obtained via a synthesis of techniques drawn from several places in his vast body of work. The line of investigation which resulted in this paper originated in our work with Shelah in [8], where I noticed that the argument in Section 4 of Chapter III in [26] which purported to extend the main coloring theorem in that section to cardinals of the form µ + for ℵ 0 = cf(µ) < µ did not work. A close study of the problem led to the isolation of the coloring theorems in [7], and in the joint paper [9] we exploited a combinatorial trick to provide a partial rescue of the result from [26]. Some years earlier, Shelah had suggested the idea of "off-center" club-guessing as a possible way of repairing the error in [26], but nothing came of the idea at the time because the combinatorics did not work as we hoped. While writing up [9], the author realized that the "offcenter" approach might be viable when combined with the combinatorics of that paper. The work in sections 3 through 9 shows that this was indeed the case -in Theorem 6 we get a version of the main result from [7] which holds for successors of singular cardinals of countable cofinality. The jump from this to getting results on simultaneous reflection of stationary sets is based on other work of Shelah -Section 2 and 3 of this paper can be viewed as teasing out additional consequences from arguments appearing in Section 3 of Chapter III from [26].

Weak saturation and indecomposability
Our goal in this section is a modest one -we will take two fairly well-known properties of ideals and show, using elementary arguments, that their conjunction is equivalent to several useful properties. The results appearing in this section have a "folklore flavor" and may be known; Shelah maps out some of the implications appearing here in Chapter III of his [26], but he stops short of proving that many of the properties he was considering are in fact equivalent. Theorem 1 at the end of this section gathers the results in a single place.
Before proving anything, we take a moment to discuss some notational conventions and expressions that will appear over and over in our results. This terminology is fairly standard, so readers familiar with such things can just skip ahead.
Definition 2.1. Let I be an ideal on κ.
(1) I * denotes the filter dual to I.
(2) Expressions of the form "ϕ holds for I-almost all α < κ" mean that the set of α < κ for which ϕ holds is in I * . If I is clear from context, we may omit explicit reference to it and say only "ϕ holds for almost all α < κ". (4) Similarly, if f and g are ordinal-valued functions with domain κ, we say for almost all α < κ. The expression f = I g is also given the obvious meaning.
Variants of the above notation should also be interpreted in the canonical fashion.
We look now at the first of two properties which form our main interest in this section.
Definition 2.2. Let I be an ideal on the 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 provides a measure of how closely the filter I * dual to I comes to being an ultrafilter. For example, I * is an ultrafilter if and only if I is weakly 2-saturated, while the ideal of bounded subsets of κ fails to be weakly κ-saturated. This concept has not been studied as systematically as its better-known relative "saturation", but we shall see that it is quite important in its own right. The following observation starts us on our way. (1) I is weakly θ-saturated.
(2) Any ⊆-increasing θ-sequence A i : i < θ of subsets of κ is eventually constant modulo I.
Proof. The proof is trivial. For example, one proves that (1) implies (2) via contradiction: If (2) fails, then we can refine A i : i < θ to a subsequence B i : i < θ with the property that B i ⊆ I B i+1 and B i+1 \ B i / ∈ I for each i < θ. The collection B i+1 \ B i : i < θ quickly leads to a contradiction of (1).
The next concept we need for our discussion is that of indecomposability, defined as follows: Definition 2.4. Let I be an ideal on κ, and let θ be a cardinal. The ideal I is said to be θ-indecomposable if whenever A i : i < θ is a θ-sequence of subsets of κ with i<θ A i / ∈ I, there is a set w ⊆ θ of cardinality less than θ with i∈w A i / ∈ I. In the case where θ is a regular cardinal, this is equivalent to the statement that the ideal I is closed under increasing union of length θ.
Indecomposability has been considered many times in the literature (see [3] and [20] for example), although one usually finds the definition phrased in terms of filters rather than ideals, and very often the authors restrict themselves to considering ultrafilters rather than the more general case. Proposition 2.5. Let I be an ideal on the cardinal κ. The following two statements are equivalent for a regular cardinal θ: (1) I is θ-decomposable (2) There is a θ-sequence A i : i < θ of I-equivalent I-positive sets with Proof. Suppose I is θ-decomposable. Since θ is regular, this means there is an increasing sequence B i : i < θ of elements of I whose union B is I-positive. If we define For the other direction, suppose we are given A i : i < θ as in (2). Let B = A 0 , and define Note that B i ∈ I as B and A i are equivalent modulo I. The sequence B i : i < θ is also increasing, and by (2.1) we have Since B / ∈ I, we conclude that I is θ-decomposable.
Our main concern in this section is a consideration of the conjunction of "θindecomposable" and "weakly θ-saturated" for a regular cardinal θ. The next proposition shows us that this combination is has some strength -it is equivalent to improvements of both Proposition 2.3 and Proposition 2.5. Proposition 2.6. Let I be an ideal on the cardinal κ. The following statements are equivalent for a regular cardinal θ.
(2) Whenever B i : i < θ is an increasing θ-sequence of subsets of κ, there is an i * such that (3) Whenever A i : i < θ is a θ-sequence of I-positive subsets of κ, we have Proof. Assume (1) holds, and let B i : i < θ be an increasing θ-sequence of subsets of κ. Since I is weakly θ-saturated, we know that this sequence is eventually constant modulo I, so fix i * < θ with the property that Let B = i<θ B i , and note that (2) is established if we can prove that B \ B i * is in I. This is done through θ-indecomposability, for B \ B i * can be expressed as the union of an increasing θ-sequence of elements of I: Next, assume that condition (2) holds for our ideal, and assume by way of contradiction that A i : i < θ is a θ-sequence of I-positive subsets of κ for which (2.3) fails. For each i < θ, define The sequence B i : i < θ is increasing, and furthermore i<θ B i = κ because (2.3) fails. By (2), there is an i * such that B i * = I κ. This is a contradiction, as A i * is I-positive and disjoint to B i * .
To finish the proof, we show that the failure of (1) implies the failure of (3). This is easily done -if I is not weakly θ-saturated than any partition of κ into θ disjoint I-positive sets will contradict (3), and if I is θ-decomposable then we contradict (3) by way of Proposition 2.5.
Notice that condition (2) says something stronger than the conclusion of Proposition 2.3 -the sequence of sets is not only eventually constant modulo I, it is the case that eventually the individual sets B i are equal to the union of the entire sequence modulo I.
Condition (3) says that any "point-< θ" collection of I-positive sets has size less than θ, for given a collection of θ (or more) I-positive sets, there is a subcollection of size θ with non-empty intersection.
Since prime ideals are weakly 2-saturated, we obtain the following (known) characterization of θ-indecomposability in the context of ultrafilters.
Corollary 2.7. Let U be an ultrafilter on some cardinal κ, and let θ be a regular cardinal. Then U is θ-decomposable if and only if it is (θ, θ)-regular.
Proof. By definition U is (θ, θ)-regular if and only if there is a family {A i : i < θ} of elements of U with the property that the intersection of any subfamily of size θ is empty. This is precisely the condition (2.3), and so the result follows immediately as the ideal dual to U is trivially weakly θ-saturated.
Our next move takes us to the realm of functions modulo ideals, on the cusp of pcf theory. Proposition 2.8. Let I be an ideal on the cardinal κ. The following two statements are equivalent for any regular cardinal θ < κ: (1) I is weakly θ-saturated and θ-indecomposable.
(2) Suppose S α : α < κ is a sequence of sets of ordinals with |S α | < θ for all α. Then any ≤-increasing θ-sequence of functions in α<κ S α is eventually constant modulo I.
Proof. Both directions make use of condition (3) in Proposition 2.6. Let us suppose S α : α < κ is as in (2), and assume by way of contradiction thath = h i : i < θ is a ≤-increasing sequence in α<κ S α that is not eventually constant modulo I. By passing to a subsequence we may assume Fix α < κ, and suppose α ∈ A i ∩ A j for some i < j < θ. Sinceh is ≤-increasing, we have Since |S α | < θ, we conclude and therefore.
Each A i was assumed to be I-positive, so we have contradicted (1). For the other direction, let us assume that (1) fails. By Proposition 2.6, we can find a sequence A i : i < θ of I-positive subsets of κ such that i<θ i≤j<θ We define a sequenceh = h i : i < θ of functions in κ Ord by the following recursion: Case 1: Initial stage We define h 0 to be identically 0.

Case 3: Limit stages
If i is a limit ordinal, then we define h i by setting h i (α) = sup{h j (α) : j < i}. Now let us define S α := {f i (α) : α < θ}. Since (2.6) holds, our construction guarantees that |S α | < θ for all α. Clearlyh is ≤-increasing, and since no A i is in I, the sequence is also not eventually constant modulo I.
The proof of the above is easily modified to yield the following slightly strengthened result: Corollary 2.9. Suppose I is weakly θ-saturated and θ-indecomposable ideal on κ for some regular cardinal θ, and let S α : α < κ be a sequence of sets of ordinals with |S α | < θ. Ifh = h β : β < τ is a ≤-increasing sequence of functions in α<κ S α with cf(τ ) ≥ θ, thenh is eventually constant modulo I. Another easy characterization obtained by slightly different methods is the following: Proposition 2.10. The following statements are equivalent for an ideal I on κ and regular θ < κ: (1) I is weakly θ-saturated and θ-indecomposable.
We prove the other direction by contrapositive -if (1) fails, then by (3) of Proposition 2.6 we can find a family A i : i < θ of I-positive subsets of κ such that Since (2.7) holds, it follows that f maps κ into θ. However, f is not bounded below θ almost everywhere, as for each β < θ, we know The proof of the above easily yields something slightly stronger: Corollary 2.11. Suppose I is a weakly θ-saturated θ-indecomposable ideal on κ for some regular θ < κ. If δ is an ordinal of cofinality θ, then any function f : κ → δ is bounded below δ almost everywhere.
We close this section by formulating a theorem summarizing the above results. Theorem 1. The following statements are equivalent for an ideal I on κ and a regular cardinal θ < κ.
(2) Whenever B i : i < θ is an increasing θ-sequence of subsets of κ, there is an i * such that (4) If S α : α < κ is a sequence of sets of ordinals with |S α | < θ for all α, then any ≤-increasing θ-sequence of functions in α<κ S α is eventually constant modulo I.

Least functions and a coloring theorem
In this section, we show that ideals I of the sort considered in Theorem 1 necessarily entail the existence of a stationary S * ⊆ κ for which strong versions of Refl(S * ) (involving simultaneous reflection of stationary subsets of S * ) hold. Once this is established, the section closes by proving a weak version of our main theorem that does not require the complicated machinery introduced later in the paper. Once again, we need to recall some terminology from the general theory of ideals.
Definition 3.1. Let I be an ideal on a cardinal κ.
The following proposition shows the relevance of the preceding definition to the properties we studied in the preceding section. Shelah (Claim III.3.2A of [26]) obtains the same conclusion from a weaker hypothesis (his proof is the natural generalization of work of Kanamori and Ketonen [14,16,17] to the context of notnecessarily-prime ideals), but we include this proof in the interest of completeness and because our assumptions simplify the argument. Proposition 3.2 (Shelah). Let I be an ideal on κ, and suppose I is θ-indecomposable and weakly θ-saturated for some regular θ < κ. Then there is a function f * : κ → κ such that (1) for each β < κ we have f * (α) > β for almost all α < κ, and (2) if g(α) < f * (α) for almost all α < κ, then there is a β < κ such that g(α) < β for almost all α < κ.
In particular, f * is a least function modulo I.
Proof. When referring to statement (1), we will be a little imprecise and say that "f * is an upper bound for the constant functions modulo I" and rely on context to make it clear to which constant functions we are referring. We will similarly refer to the conclusion of (2) by saying "g is bounded modulo I". • h is an upper bound for the constant functions modulo I, Proof. By our assumptions, we can find a function g : κ → Ord such that g < I f , but g is not bounded modulo I. We can freely modify g on sets in I, so we may as well assume g(α) ≤ f (α) for all α < κ.
For each β < κ, let us define Since the sequence A β : β < κ is ⊆-increasing, we can apply Proposition 2.3 and conclude that there is a β * such that Note that the complement of A β * is not in I because we assumed g is not bounded modulo I. Furthermore, by (3.1) we see that Thus, if we define we have what we need. (Notice that we only needed the weak θ-saturation for the proof of this lemma.) Assuming by way of contradiction that there is no function f * answering conditions (1) and (2), we use the preceding lemma to build build a I -decreasing sequence of functions f ξ : ξ < θ , each of which is an upper bound for the constant functions modulo I, using the following inductive recipe: Let f 0 : κ → κ be the identity function. Since I contains all bounded subsets of κ, it is clear that f 0 is an upper bound for the constant functions modulo I.
One should view h β as a "projection" of the function that is constant with value β up into the product α<κ S α . Our choice of f 0 guarantees that h β (α) < κ for almost all α < κ. Finally, we note that the sequence h β : β < κ is ≤-increasing, and so an application of Proposition 2.8 tells us that there is an ordinal β(ξ) < κ such that h β = I h β(ξ) whenever β(ξ) ≤ β < κ. We now define Notice that f ξ is an upper bound for the constant functions modulo I, and moreover for each ζ < ξ, if β(ξ) ≤ f ζ (α) (something that happens for almost all α < κ), then it must be the case that Thus f ξ ≤ I f ζ for every ζ < ξ. Since ξ is a limit ordinal, our actions at successor stages guarantee that in fact f ξ I f ζ whenever ζ < ξ.
The preceding construction generates a sequence of functions f ζ : ζ < θ as well as ordinals β(ξ) for each limit ordinal ξ < θ. Let us define β * := sup{β(ξ(ǫ)) : ǫ < θ}, and for each limit ξ < θ, we set First, note that each A ξ is I-positive, as f ξ+1 I f ξ and f ξ+1 is an upper bound for the constant functions modulo I. More importantly, we have the following claim: Thus, we have and the proof is complete.
The above claim makes it clear that no α < κ can belong to infinitely many of the sets A ξ . In particular, this collection of I-positive sets contradicts part (3) of Theorem 1, and we are done.
Our next move is to show that the conjunction of θ-indecomposability and weak θ-saturation has strong consequences for stationary reflection. The following definition will allow us to state some conclusions more precisely: (1) Comp(I) is the least cardinal τ for which I is τ -complete, that is, for which I is closed under unions of fewer than τ sets.
(2) Wsat(I) is the least cardinal θ for which I is weakly θ-saturated.
(4) S * (I) = {α < κ : Wsat(I) ≤ cf(α) < α and cf(α) ∈ Indec(I)} Consider the set S * (I) defined above for a moment. Notice that it is non-empty if and only if there is a regular θ < κ for which I is both weakly θ-saturated and θ-indecomposable. Also, if S * (I) is non-empty, then it is stationary. The omission of regular cardinals from S * (I) is only relevant if κ happens to be Mahlo; we define the set this way so that Theorem 2 applies uniformly to any cardinal.
Theorem 2. Let I be an ideal on the cardinal κ. If there is a regular cardinal θ < κ such that I is weakly θ-saturated and θ-indecomposable, then for almost all α < κ. In particular, (4) S * (I) is a stationary subset of κ for which Refl(< Comp(T ), S * (I)) holds.
Proof. We have already remarked that (1) is a consequence of the given hypotheses, and (2) is the conclusion of Proposition 3.2. It should also be clear that (4) follows from (3), so we will spend our time establishing (3).
Let S be a stationary subset of S * (I). Since (3) remains true if we establish its conclusion for a stationary subset of S, we can take advantage of the fact that cofinality function is regressive on S * (I) and assume that S is a subset of S κ τ for some τ . Notice that I is weakly τ -saturated and τ -indecomposable because of the definition of S * (I).
It suffices to prove that whenever we are given a sequence C α : α < κ, α limit} with each C α closed unbounded in α, we have Given β < κ, let us define f β : κ → κ by Bear in mind that f β (α) < f * (α) for almost all α < κ, and so our choice of f * implies f β is bounded almost everywhere. Thus, there is a function g : κ → κ with the property that for any β < κ, Let E be the closed unbounded subset of κ consisting of those ordinals closed under the function g, and fix δ ∈ E ∩ S. Since δ is closed under g, we know Now we define another function h by The function h maps κ to δ. Since cf(δ) = τ and I is τ -indecomposable, we conclude from Corollary 2.11 that there is a β < δ for which Thus, almost all α < κ satisfying the following: We finish the proof of (3.4) by establishing that δ ∈ C f * (α) for all α < κ which satisfy both of these statements. Given such an α, assume by way of contradiction that δ / ∈ C f * (α) . On one hand we must have while on the other, we have This latter equation implies and clearly (3.9) and (3.11) contradict each other.
After we noticed the above theorem, we discovered that Shelah uses essentially the same argument in a different context -it appears tucked into "Proof of 3.3 in Case β, Subcase (a): Second Proof" on page 149 of [26] in a result having to do with weakly inaccessible cardinals. Theorem 2 still appears to be new -one can view it as "what Shelah's argument really shows".
To see the power of the preceding result, we include the following theorem. This result is essentially a special case of our main theorem. It arises from combining the preceding theorem with one of the main results from [9], and its proof serves as a prototype for the argument we employ in Section 10.
Proof. Corollary 5.2 of [9] (which itself relies on results in [7]) tells us that under our assumption, there is an ideal I on µ + such that • I is τ -indecomposable for all regular τ with cf(µ) < τ < µ, and • I is weakly θ-saturated for some θ < µ.
Since µ + is certainly not a Mahlo cardinal, it follows from the above that S * (I) is equal to S µ + θ modulo the non-stationary ideal, and therefore the conclusion follows from Theorem 2.
We remark that in the case where cf(µ) is uncountable, we can use results from [7] to get the same conclusion from the failure of Pr 1 (µ + , µ + , µ + , cf(µ)); we will have more to say about this later.

Off-center club guessing
Our focus now shifts away from the general theory of ideals to questions involving club-guessing. The proof of Theorem 3 turned on properties of an ideal I whose existence will seem somewhat mysterious to those not familiar with [7] and [9]. The ideal referenced in the proof of Theorem 3 is related to club-guessing, and this explains why we turn our attention to this matter.
In general, a prototypical club-guessing theorem provides one with a stationary subset S of some cardinal κ, and a sequence C δ : δ ∈ S (called an S-club sequence) such that • C δ is closed and unbounded in δ for each δ ∈ S, and • for every closed unbounded E ⊆ κ, there are "many" δ ∈ S for which E ∩C δ is "large".
One can require various conditions on the sets C δ , as well as varying the specific meaning of "many" and "large". The most well-known club-guessing theorems give us (in certain circumstances) an S-club sequence C δ : δ ∈ S for which otp(C δ ) = cf(δ), and such that for each closed unbounded E ⊆ κ, there are stationarily many δ ∈ S for which C δ ⊆ E. It should be clear that such theorems fit our prototype.
The techniques of [26,8,7,9], however, require a special sort of club-guessing that is, in a sense, both weaker and stronger than the standard results. We will give a rough explanation of this, but we need to enlarge our vocabulary a little bit first: • nacc(C) = C \ acc(C).
Here "acc" stands for "accumulation points" and "nacc" for non-accumulation points. Finally, • if α ∈ nacc(C), then we define Gap(α, C), the gap in C determined by α, by Returning now to our discussion, the earlier papers needed results stating that if µ is singular and S ⊆ S µ + cf(µ) , then there is an S-club sequence C δ : δ ∈ S such that for every closed unbounded E ⊆ µ + , there are stationarily many δ ∈ S such that . This type of guessing is weaker than the standard sort in that we aren't requiring C δ to be a subset E, but it is also stronger in that we do demand that C δ ∩ E contains lots of ordinals of large cofinality. Without going into specifics, if the cofinality of µ is uncountable, then we can prove the existence of extremely nice club-guessing sequences with the properties we want (these are the S-good pairs of [7]) -sequences so nice that they can be used to generate colorings of [µ + ] 2 . The club-guessing result we used in that paper (due originally to Shelah, but a proof can be found in [9]) does not seem to generalize to the case where µ has countable cofinality. The paper [9] obtains a weaker result for that case, and the conclusions drawn there are correspondingly weaker -this is why Theorem 3 refers to square-brackets relations instead of the stronger Pr 1 (µ + , µ + , µ + , cf(µ)), and it also explains the existence of the current work.
One of the main goals of this paper is to remedy the situation by moving the club-guessing "off-center" -instead of focusing on subsets of S µ + cf(µ) , we look at stationary subsets consisting of ordinals of larger cofinality. This is done at a price, for the sets C δ we construct are necessarily much more complex. Despite this added complexity, we are forced to keep tight control over their structure because we still want to be able to connect these club-guessing sequences with the existence of complicated colorings.
This section presents the club-guessing result alluded to in the preceding paragraph. The theorem is established by modifying some club-guessing arguments from [9] to this new context. The main difficulty in this generalization has to do with the proliferation of parameters, so we start with a list of our main assumptions and notation: • λ = µ + for µ a singular cardinal • κ = cf(µ) • κ < σ = cf(σ) < µ • S is a stationary subset of S λ σ • µ i : i < κ is a continuous increasing sequence of cardinals cofinal in µ, • c δ : δ ∈ S is a family of functions such that c δ is an increasing and continuous function from σ onto a cofinal subset of δ, and for every closed unbounded E ⊆ λ, there are stationarily many δ ∈ S for which ran(c δ ) ⊆ E. • for α < σ, I δ α denotes the half-open interval (c δ (α), c δ (α + 1)]. Note that if we were to define C δ to be ran(c δ ) for δ ∈ S, then we would end up with a standard sort of club-guessing sequence -for every closed unbounded E ⊆ λ, there would be stationarily many δ ∈ S for which C δ ⊆ E.
The next definition captures some standard ideas from proofs of club-guessing; in some cases we have chosen to use more descriptive names (mostly due to Kojman [18]) for these operations than the terminology prevalent in [26]. Furthermore, if C and E are both subsets of some cardinal λ and e α : α < λ is a C-sequence, then for each α ∈ nacc(C) ∩ acc(E), we define The names help one to visualize what the operations do -Drop(C, E) is the result of "dropping" C into the set E, while "Fill" gives us a reasonably canonical way of turning non-accumulation points into accumulation points. Given these two operations, we are now in a position to prove the following club-guessing theorem: Theorem 4. There is an S-club systemC = C δ : δ ∈ S such that (1) ran(c δ ) ⊆ C δ (2) for each ǫ < σ and i < κ, C δ ∩ I δ ǫ·κ+i has cardinality ≤ µ + i (3) if α ∈ nacc(C δ ) ∩ I δ ǫ·κ+i , then cf(α) > µ + i (4) for every club E ⊆ λ, for stationarily many δ ∈ S, for every ǫ < σ and i < κ, E ∩ nacc(C δ ) ∩ I δ ǫ·κ+i is non-empty. Proof. We start by simplifying our goal somewhat, by noting that it suffices to produceC satisfying (1), (2), and the following modified version of (4): (4) ′ for every club E ⊆ λ, for stationarily many δ ∈ S, for every ǫ < σ and i < κ, E ∩ nacc(C δ ) ∩ I δ ǫ·κ+i contains an ordinal of cofinality > µ + i . Why does this suffice? Given such an S-club system, we simply throw away those members of nacc(C δ ) whose cofinalities are too small to obtain something satisfying (3), and note that the club-guessing properties we need are not harmed by this pruning.
Moving on to the proof, let us assume by way of contradiction that there is no S-club systemC satisfying (1), (2), and (4) ′ . Our aim is to exploit this assumption in order to construct a certain sequence C ζ : ζ < σ + of S-club systems which will then be used to produce a countable decreasing sequence of ordinals.
Let us agree to say that an S-club system satisfies the structural requirements of Theorem 4 if conditions (1) and (2) of the conclusion of the theorem hold. We will define objects E ζ andC ζ = C ζ δ : δ ∈ S by induction on ζ < σ + . Each E ζ will be closed and unbounded in λ, while eachC ζ will be an S-club system satisfying the structural requirements of Theorem 4. Our convention is that "stage ζ" in our construction refers to the process of building E ζ+1 andC ζ+1 from E ζ andC ζ . Our initial set-up is to take E 0 = λ and C 0 δ = ran(c δ ) for each δ ∈ S. Stage ζ: Defining E ζ+1 andC ζ+1 We assume that our construction furnishes us with an S-club systemC ζ satisfying the structural requirements of Theorem 4. Our assumption is that the theorem fails, and so there are closed unbounded subsets E 0 ζ and E 1 ζ of λ such that for each δ ∈ E 0 ζ , we can find ǫ < σ and i < κ such that . The definition ofC ζ+1 will take a bit more effort. Let us agree to call an ordinal δ ∈ S active at stage ζ if C 0 δ ⊆ acc(E ζ+1 ). Our choice of c δ : δ ∈ S ensures that at any stage, the set of active δ is a stationary subset of S. If δ ∈ S is inactive at stage ζ, then we simply define C ζ+1 δ to be C ζ δ and do nothing. On the other hand, if δ is active at stage ζ, then δ must be in E 0 ζ and therefore we can find a least a(δ, ζ) < σ such that Our construction of C ζ+1 δ will modify C ζ δ only on the interval I(δ, ζ) -everything else will be left untouched.
We now define C ζ+1 δ in piecewise fashion: The S-club systemC ζ+1 δ satisfies the structural requirements of our theorem, and so the construction can continue.
We still need to describe how to obtainC ζ and E ζ when ζ is a limit. Our construction defines and for each δ ∈ S, we define C ζ δ to be the closure in δ of {α < δ : α ∈ C ξ δ for all sufficiently large ξ < ζ}. Note that C ζ δ is closed in δ, and it is unbounded in δ as it contains C 0 δ . Elementary cardinal arithmetic implies thatC ζ satisfies the structural requirements of Theorem 4, and so our construction can continue.
To this point in the proof, we have used the failure of Theorem 4 to produce a sequence C ζ : ζ < σ + of S-club systems; our task is to show that this leads to a contradiction.
Both δ and β n are in E ζn+1 , so in particular we know δ ∈ E 0 ζn and β n ∈ E 1 ζn . In addition, β n must lie in nacc(C ζn δ ) because β * < β n . By (4.4), we conclude that cf(β n ) ≤ µ + i(δ,ζn) . We have assumed in our case hypothesis that β n = sup(E ζn+1 ∩ β n ) and this guarantees that β n is an element of D ζn δ . Since β n is also in nacc(C ζn δ ), the set D ζn δ cannot pick up any new elements between β * and β n and hence (4.12) β n = min(D ζn δ \ β * ) > β * . Thus, β n is in nacc(D ζn ) and we see that β n needs attention during the construction of C ζn+1 δ .
In summary, if the conclusion of our theorem fails, then our construction generates an infinite decreasing sequence of ordinals. This is absurd, and so the theorem is established.

Organizational interlude
Our goal in this section is to lay a good foundation before proceeding to the proofs of our main theorems. A good deal of the difficulty in these proofs lies in the fact that we require so many different objects to push the argument through; we make an attempt at organizing notation and providing a clear picture of our assumptions before proceeding. In addition, we focus only on the case where cf(µ) = ℵ 0 , as uncountable cofinalities can be handled by earlier work. We first fix our names for the various cardinals that are important for our theorems: • λ = µ + for µ singular of cofinality ℵ 0 .
If δ ∈ S is fixed and clear from context (as is usually the case), then we will write I(ǫ, m) instead of I(δ, ǫ, m). It is crucial for the reader to have a good picture of the structure of the objects described above. Our notation is intended to describe something fairly simple: given δ ∈ S, we use c 0 δ (or rather, the range of c 0 δ ) essentially to divide the ordinals less than δ into σ blocks, each of which is further divided into ω pieces. We say "essentially", because there are a few ordinals left out for technical reasons, but the reader will be well-served by thinking of the interval I(δ, ǫ, m) as "the mth piece in block ǫ built using C δ ". The following observation tells the complete story.
Our next task is to connect our club-guessing sequence to some ideals introduced by Shelah in [26]. We have already discussed club-guessing in general terms at the start of the previous section; what follows is a deeper discussion the particular case of interest to us.

Shelah uses the notation J b[µ]
C δ for the ideal described above; we have gone with something a little simpler. These ideals I δ are natural in our given context -for example, the following two propositions show us that the ideals are compatible with the interval structure we have imposed on the objects C δ Proposition 5.3. Suppose δ ∈ S. If ǫ * < σ and m * < ω are fixed, then I δ -almost members of C δ are in nacc(C δ ) ∩ I(ǫ, m) for some ǫ ≥ ǫ * and m ≥ m * .
Proof. This is clear from the definition of I δ .
Proposition 5.4. If E is a closed unbounded subset of λ, then there are stationarily many δ for which E ∩ C δ / ∈ I δ .
Proof. Our assumptions give us a stationary set of δ such that E ∩nacc(C δ )∩I(ǫ, m) is non-empty for each ǫ < σ and m < ω, so we are done if we verify that E ∩C δ / ∈ I δ for each such δ. This is easily done -given α < µ and β < δ, we need to show that E ∩ C δ contains a member of nacc(C δ ) larger than α and whose cofinality is greater than β. Choose ǫ < σ so large that α < c 0 δ (ω · ǫ) and choose m so large that β < µ m . We know E ∩ nacc(C δ ) ∩ I(ǫ, m) is non-empty, and any ordinal in this intersection is necessarily a member of E ∩ nacc(C δ ) larger than α and of cofinality greater than β.
Given the preceding proposition, we can now bring in a certain club-guessing ideal which will be woven into all of our most important results. The following definition makes sense in a more general context than that which we are considering (Chapter III of [26], for example), but we will deal only with the particular case of interest to us. Definition 5.5. LetĪ = I δ : δ ∈ S be as in Definition 5.2. The ideal id p (C,Ī) is defined by A ∈ id p (C,Ī) if there is a closed unbounded E ⊆ λ such that The preceding definition is robust under slight modifications. For example, we get the same ideal by requiring A ∩ E ∩ C δ to be in I δ for all but non-stationarily many δ ∈ S ∩ E. Note as well that the restriction to δ ∈ S ∩ E is not important -if δ ∈ S \ E, then E ∩ C δ must be bounded and therefore E ∩ C δ ∈ I δ . Roughly speaking, a set A ⊆ λ is in id p (C,Ī) ifC fails to "guess" E ∩ A in the sense of Proposition 5.4. Note that in light of Proposition 5.4, the ideal id p (C,Ī) is a proper ideal on λ extending the non-stationary ideal.
The following fact is a specific instance of Observation 3.2 in Chapter III of Shelah -it is an elementary result establishing indecomposability properties for the club-guessing ideal under consideration. We include the proof because the result is crucial piece of our main theorem.
Proof. Let τ = σ be an uncountable regular cardinal less than µ. Given δ ∈ S, we first establish that I δ is τ -indecomposable, so let A i : i < τ be an increasing sequence of subsets of C δ , each of which is in I δ . The collection of sets of the form (5.1) is closed under finite unions, and since these sets generate I δ , it follows that each set in the ideal is covered by a single set of that form. Thus, we may define α i to be the least α < µ for which there is a β < δ such that A i ⊆ {γ ∈ C δ : γ ∈ acc(C δ ) or cf(γ) < α or γ < β}.
Since the sequence A i : i < τ is increasing, it is clear that α i : i < τ is a non-decreasing sequence of ordinals less than µ. Since cf(µ) = ℵ 0 and τ is an uncountable regular cardinal, it follows this sequence is bounded by some ordinal α * < µ.
Next, we let β i be the least ordinal β < δ for which Such a β always exists because of our choice of α * , and since A i : i < τ is increasing, the sequence β i : i < τ is non-decreasing. Since τ = σ = cf(δ), there is an ordinal β * < δ greater than all β i . Thus, we conclude and hence this union is in I δ .
The rest of the argument consists in noting that id p (C,Ī) inherits the indecomposability properties of the ideals I δ . To see this, let A i : i < τ be an increasing family of subsets of λ, each of which is in id p (C,Ī). Set A equal to the union of these sets, and suppose by way of contradiction that A / ∈ id p (C,Ī).
If we let E be the intersection of all the sets E i , then it is clear that E is closed and unbounded in λ. We have assumed that A / ∈ id p (C,Ī), and thus we can fix an ordinal δ ∈ S ∩ E such that Define It should be clear that the sequence B i : i < τ is a ⊆-increasing sequence of subsets of C δ , and since E ⊆ E i , we know B i ∈ I δ by (5.3). Now I δ is τ -indecomposable, and so This is a contradiction, though, because we also know which is not in I δ by (5.4). We conclude that A must be in id p (C,Ī), as required.
Now that we have established some basic results about our club-guessing sequence, our concern shifts to seeing how well some of the techniques used in [9] can be adapted to this context. We begin with an observation that the interval structure we have placed on our sets C δ gives us a natural way of writing each C δ as a countable increasing union of approximations: Definition 5.7. Given δ ∈ S and m < ω, we define In more descriptive language, C δ [m] consists of ran(c 0 δ ) together with the parts of C δ lying in the first m + 1 pieces of each of the σ blocks built using C δ . We note the following easy facts about these objects: Our next move is implement "Shelah's ladder-swallowing trick" (a crucial ingredient of the proofs in [7] and [9]) in this new context. We take the following definition from [9]. Definition 5.9. A generalized C-sequence on λ is a family e m α : α < λ, m < ω such that for each α < λ and m < ω, • e m α is closed unbounded in α, and • e m α ⊆ e m+1 α .
We now use our collection of objects C δ [m] in order to construct a very special generalized C-sequence.  (1) and (2) in the preceding lemma are crucial ingredients in the proof of our main result. The name "ladder-swallowing trick" comes from condition (2), which tells us that the generalized C-sequence we have built "swallows" the objects C δ [n] in a natural way. These sorts of requirements probably seem quite unmotivated at this point, but we note that it is precisely this "swallowing" behavior that allows us to lift some of Todorčević's minimal walks machinery to successors of singular cardinals without the need for strong assumptions like (a technique first pioneered by Shelah in Chapter III of [26], as well as [8]).

Minimal walks and a preliminary result
We leave club-guessing behind for a bit, and turn now to Todorčević's technique of minimal walks as it relates to our specific generalized C-sequence. The following definition presents the notation we use to keep track of everything. Definition 6.1. Given ordinals α < β < λ and m < ω, we define Next, we define ρ m 2 (α, β) = least i for which β m i (α, β) = α. For each m < ω and α < β < λ, the sequence β m i : i < ρ m 2 (α, β) is the minimal walk from β to α using the C-systemē m in the sense of [29]. We will abbreviate this somewhat, and call this the m-walk from β to α. The use of ρ 2 to stand for the length of such a walk is due to Todorčević. Minimal walks in the context of generalized C-sequences were introduced in [9]; our notation is a variant of the notation in that paper. Now that we have the definition of m-walk at our disposal, the statement of the following theorem makes sense. This theorem illustrates how the structure of our generalized C-system causes lets us establish connection between the m-walks defined above, and the club-guessing ideal id p (C,Ī) discussed in the preceding section. This theorem will quickly be superseded by results in the next section, but it represents an important step on the way to our main theorem.
The proof of Theorem 5 will fill the rest of this section. We begin following lemma and corollary, which are translations into the context of generalized C-sequences of some standard facts about minimal walks. Here we see our assumption that elements of S have uncountable cofinality starts to become relevant. Lemma 6.2. Suppose δ < β < λ and cf(δ) > ℵ 0 . There is an ordinal γ * (δ, β) < δ such that β) . Since this latter set is closed, it follows that sup(δ ∩ e m β m i (δ,β) ) < δ, and if we define The following corollary isolates the importance of γ * (δ, β) -if α is any ordinal between γ * (δ, β) and δ, then for any m the m-walk from β to α agrees with the m-walk from β to δ until the penultimate step of the latter walk. This is a familiar pattern of argument invented by Todorčević. Corollary 6.3. If δ < β < λ and cf(δ) > ℵ 0 , then for any m < ω and i < ρ m 2 (δ, β) we have γ * (δ, β) < α < δ =⇒ β m i (α, β) = β m i (δ, β). Proof. The proof is a straightforward induction on i < ρ m 2 (δ, β) − 1 using the definition β m i (α, β).
The next lemma takes advantage of the fact that for each α, e m α : m < ω forms an increasing family of sets. Lemma 6.4. If α < β < λ, then there is an m(α, β) < ω such that for all m ≥ m(α, β). Moreover, Proof. This follows easily by induction once we note that for any ordinals α < β, the sequence min(e m β \α) : m < ω is non-increasing and hence eventually constant.
Armed with the last lemma, we are in a position to define another parameter needed for our construction. Definition 6.5. Given α < β < λ, define In plain language, we know that the various m-walks from β down to α are in agreement for all m ≥ m(α, β), and so γ(α, β) is simply the last ordinal visited by all of these walks before they arrive at their destination α. The last two parameters we need for the proof of Theorem 5 are given in the following definition; note that we restrict the definition to the case where δ ∈ S. Definition 6.6. Suppose δ ∈ S and δ < β < λ. We define where c 0 δ and µ m : m < ω are as in our list of assumptions. The following proposition captures a few facts about the how the various objects we have been considering interact; part (3) of the proposition has particular importance for us. Proposition 6.7. Suppose δ ∈ S and δ < β < λ.
Condition (2) gives us some more information -it claims that some of the structure of C δ survives in e m γ(δ,β) , and it is at this point that we cash in some of our assumptions for the first time (in particular, it will become clear why we work with the somewhat awkwardly defined m * (δ, β) instead of m(δ, β)). Given β * in nacc(C δ ) ∩ I(ǫ, m), we know that β * ∈ e m γ(δ,β) by (1). Condition (1) of Lemma 5.10 taken with the definition of m * (δ, β) tells us e m γ(δ,β) ≤ cf(γ(δ, β)) + µ + m ≤ µ + m .
Proof. This is an immediate consequence of Proposition 6.7.
The following claim is where the closed unbounded set E starts to become relevant.
The proof of Theorem 5 is almost complete. Given δ ∈ T , our work provides us with ǫ * < σ and m * < ω as in Claim 3. Let us define B δ := nacc(C δ ) ∩ {I(ǫ, m) : ǫ * ≤ ǫ < σ, m * ≤ m < ω}. Proposition 5.3 tells us that almost all (in the sense of I δ ) members of C δ are in B δ , while Claim 3 establishes that for any β * ∈ E ∩ B δ , there are α < β < λ and m < ω such that the m-walk from ξ to ζ passes through β * for all ζ ∈ t α and ξ ∈ t β . Thus, if we define the proof Theorem 5 will be finished if we can show A ∈ id p (C,Ī).
To establish this, we show (6.11) A ∩ E ∩ C δ ∈ I δ for all δ ∈ S.
Clearly we need only worry about those δ for which E ∩ C δ / ∈ I δ . Given such a δ, we have (6.12) A ∩ E ∩ B δ = ∅ by the definition of A. If A ∩ E ∩ C δ were I δ -positive, then this set would have non-empty intersection with B δ , as the complement of B δ is in I δ . This cannot be the case, as we would contradict (6.12). Thus, A ∈ id p (C,Ī) and the proof of Theorem 5 is complete.

Scales and elementary submodels
One of our assumptions in the last two sections is that we have at our disposal a fixed increasing sequence of regular cardinals µ n : n < ω that is cofinal in µ. This parameter was utilized in the definition of m * , but otherwise it has been in the background. In the upcoming work, we are going to need to assume that this sequence of regular cardinals carries a scale, and since our we are going to need to use quite a bit of scale combinatorics, it seems reasonable to devote some time toward fixing notation and reviewing what we need.
Given an increasing sequence µ as above, if there is a f such that ( µ, f ) is a scale, then say that µ carries a scale.
Since we have been working in the situation where the singular cardinal µ has countable cofinality, we will deal only with that special case in this section; we refer the reader to [7] or [6] for more information on these matters. Thus, from now on we assume • µ n : n < ω is an increasing sequence of regular cardinals such that µ n : n < ω is cofinal in µ, σ < µ 0 , and µ n : n < ω carries a scale f α : α < λ . That such sequences exist is a very non-trivial result of Shelah (Chapter II of [26]). Interested readers can also consult [6] or [4] for expository treatments of this subject.
The function Γ defined below is a standard combinatorial tool associated with scales; again, we tailor the definition to the context at hand. Definition 7.2. For α < β < λ, we define Γ(α, β) by For convenience, we say Γ(α, α) = ∞.
The function Γ has many nice properties -it witnesses Pr 1 (µ + , µ + , cf(µ), cf(µ)) (Conclusion 1 on page 67 of [26]), for example. We shall not make use of this fact directly, but it is certainly in the background throughout our arguments.
The next lemma is a special case of Lemma 7 in [7]. We remind the reader that notation of the form "(∃ * β < λ)ψ(β)" means {β < λ : ψ(β) holds} is unbounded below λ, while "(∀ * β < λ)ψ(β)" means that {β < λ : ψ(β) fails} is bounded below λ. Lemma 7.3. There is a closed unbounded C ⊆ λ such that the following holds for every β ∈ C: Our conventions regarding elementary submodels are standard. We assume that χ is a sufficiently large regular cardinal and let A denote the structure H(χ), ∈ , < χ where H(χ) is the collection of sets hereditarily of cardinality less than χ, and < χ is some well-order of H(χ). The use of < χ means that our structure A has definable Skolem functions, and we obtain the set of Skolem terms for A by closing the collection of Skolem functions under composition. The set Sk A (B) is an elementary substructure of A, and it is the smallest such structure containing every element of B. The following technical lemma is due originally to Baumgartner [1]; it is a fact that is quite useful in proving things about the function Γ associated with a given scale. Again, we refer the reader to [7] or [6] for a proof. As a corollary to the above, we can deduce an important fact about characteristic functions of models. Once again, the following definition is but a special case of a more general definition. In the situation of Definition 7.6, it is clear that Ch M is an element of the product n<ω µ n , and furthermore, Ch M (n) = sup(M ∩ µ n ) for all sufficiently large n < ω. We can now see that the following corollary follows immediately from Lemma 7.5.
Corollary 7.7. Let M be as in Definition 7.6. If n * < ω and we define N to be Sk A (M ∪ µ n * ), then We end this section with one more handy bit of terminology due to Shelah [24].
, and (4) M i ∩ λ is a proper initial segment of λ. If x ∈ H(χ), then we say that M is a λ-approximating sequence over x if x ∈ M 0 .
Note that if M is a λ-approximating sequence and λ = µ + , then µ + 1 ⊆ M 0 because of condition (4) and the fact that µ is an element of each M i .

Defining the coloring
In this section, we will mix ideas from [8] and [7] together with the proof of Theorem 5 in order to prove a coloring theorem in ZFC. Our assumptions and notation are as in the previous two sections, and our goal is to obtain the following theorem.
The proof of Theorem 6 makes use of the function Γ defined from the scale f α : α < λ . We begin by defining a sequence c m : m < ω of functions coloring the pairs from λ. Definition 8.1. Given α < β < λ and m < ω, we define . In English, the value of of c m (α, β) is the first place on the m-walk from β down to α where "Γ changes". Except for the parameter m, this is the same coloring we used in [7].
The following lemma contains the heart of the proof of Theorem 6; it shows that the sequence of colorings c m : m < ω has many of the properties we need.
Lemma 8.2. If t α : α < λ is a pairwise disjoint collection of finite subsets of λ, then for id p (C,Ī)-almost all β * < λ, there are an m < ω and β > β * such that Proof. Let t α : α < λ be given. Clearly we may assume α ≤ min(t α ) for all α, as we can pass to a subsequence of cardinality λ with no loss of generality. Define A to be the set of β * < λ for which it is impossible to find an m < ω and β > β * with the required properties. Assume by way of contradiction that Let x = {C,ē, S, µ, λ, S, µ i : i < ω , f α : α < λ , t α : α < λ } -all the parameters needed to define the sequence c m : m < ω together with the sequence of finite sets under consideration -and let M = M i : i < λ be a λ-approximating sequence over x. The set is closed and unbounded in λ, so by our assumption (8.3), there is a δ ∈ S for which (8.4) A ∩ E ∩ C δ / ∈ I δ .
The conclusion of Corollary 6.8 tells us that if β * ∈ E ∩ nacc(C δ ) ∩ I(ǫ, m) for some ǫ ≥ ǫ * and m ≥ m * , then there is an ordinal γ * < β * such that whenever t α is contained in the interval (γ * , β * ), for every ζ ∈ t α and ξ ∈ t β we have β m k (ζ, ξ) = β m k (δ, ξ) for all k < ρ m 2 (δ, ξ). Proposition 5.3 tells us that I δ -almost all elements of C δ lie in In particular, we can find β * ∈ A∩E ∩C δ such that β * ∈ I(ǫ, m) for some ǫ ≥ ǫ * and m ≥ m * , and hence for which is an ordinal γ * < β * with the properties promised in the discussion surrounding (8.5) and (8.6). Our goal is to get a contradiction by establishing that m and β witness that this β * is not an element of A.
We observe that since β * = sup(M β * ∩ λ), β * must be a member of every closed unbounded subset of λ that is itself an element of M β * . We will shortly make use of this in the context of Lemma 7.3, but for now we observe the following simple fact: For each α < λ, we define a function f min α as follows: f min α (i) = min{f ζ (i) : ζ ∈ t α }. Since each t α is finite, it follows that (∀ * i < ω)[f min α (i) = f min(tα) (i)] and therefore f min α : α < λ is a scale. Since this new scale is definable from parameters available in M 0 , it itself must be a member of M 0 as well.
As observed earlier, β * is in every closed unbounded subset of λ that is also an element of M β * . Thus, an application of Lemma 7.3 inside of M β * to the scale f min α : α < λ tells us (7.2) holds for β * . In particular, we can fix i 0 < ω so that whenever i 0 ≤ i < ω, it is the case that . The next part of our argument is going to require some Skolem hull arguments. We start by defining Note that x∪{β * } ∈ M β * +1 (as everything except β * is already in M 0 ) and therefore M ∈ M δ as it is definable in M δ by taking the Skolem hull of x ∪ {β * } inside the structure M β * +1 . Since M is countable and µ 0 is not, it follows that and therefore (8.9) f (i) ≤ Ch M (i) for all f ∈ M ∩ n<ω µ n and i < ω.
Proof of Theorem 6. Suppose β * is as in the conclusion of Lemma 8.2, and fix m and β for which (8.2) is true. We can find an i < ω sufficiently large so that • p(i) = m, and For such α, we have (8.19) c(ζ, ξ) = β * for all ζ ∈ t α and ξ ∈ t β , just as required by Theorem 6.
We finish this section with a corollary whose proof involves applying a well-known trick to the coloring from Theorem 6. Corollary 8.5. Let θ ≤ λ be a cardinal. If id p (C,Ī) is not weakly θ-saturated, then Pr 1 (λ, λ, θ, ℵ 0 ) holds.
In light of the above corollary, we see that if id p (C,Ī) is not weakly µ-saturated, then Pr 1 (µ + , µ + , µ, ℵ 0 ) holds. A similar situation occurred in [8] and [7], and in those two cases we were able to improve things to a coloring with µ + colors and obtain Pr 1 (µ + , µ + , µ + , cf(µ)). In this paper, however, the use of m-walks brings in an extra parameter that seemingly prevents the proofs from the earlier papers from being carried out. We are still able to use a "stepping-up argument" to get an upgrade to λ colors, but we pay a price in that the resulting coloring is defined on triples instead of pairs. Theorem 7. If id p (C,Ī) is not weakly µ-saturated, then there exists a coloring d : [λ] 3 → λ such that whenever t α : α < λ is a pairwise disjoint collection of finite subsets of λ, and ς < λ, we can find α < β < γ such that Proof. Our main ingredients in the definition of d are the coloring c from Theorem 6, and the coloring c * from Corollary 8.5. We also fix functions g β mapping β onto µ whenever µ ≤ β < λ. The function d : [λ] 3 → λ is defined by Suppose now that t α : α < λ is a pairwise disjoint family of finite subsets of λ. Without loss of generality, we assume α ≤ min(t α ) and max(t α ) < min(t β ) whenever α < β; this can easily be arranged by passing to a subfamily of size λ, and such a move does not interfere with our conclusion. Lemma 8.2 and the proof of Theorem 6 tells us that for id p (C,Ī)-almost all η < λ, there is value h(η) > η such that Since id p (C,Ī) contains all the non-stationary subsets of λ, Fodor's Lemma implies that there is a single ι < µ such that for stationarily many η < λ, we have both (8.22) and that g η (ι) is defined and equal to ς. Let T denote the set of all η < λ satisfying the above. By tossing away a nonstationary subset of T if necessary, we can assume that T consists of limit ordinals, and that t h(η * ) ⊆ η whenever η * < η in T . Given this, if we define then the resulting family s η : η ∈ T is a pairwise disjoint collection of finite subsets of λ.

Conclusions
In this final section, we give a proof of our main theorem. Our goal is to combine Theorem 2 with the coloring theorems from the preceding section (in the case where our singular cardinal has countable cofinality) and from [7] (in the case where the cofinality is uncountable). We will dispense with the assumptions that have been in force for the past few sections, in order to state things in full generality. We begin with a short summary of the main results of [7].
What about the case where cf(µ) = ℵ 0 ? In this case, our assumptions give us a finite sequence S i : i < n of stationary subsets of S µ + ≥θ which fail to reflect simultaneously. Without loss of generality, we can assume that S i ⊆ S µ + τi for some regular cardinal τ i . This allows us to choose a regular cardinal σ such that max{τ i : i < n} < σ < µ.
Let S = S µ + σ . An application of Theorem 4 gives us an S-club systemC for which Theorem 6 holds. The corresponding ideal id p (C,Ī) is τ i -indecomposable for i < n by Proposition 5.6, and so Theorem 2 lets us conclude that id p (C,Ī) cannot be weakly θ-saturated -it it were, then our sets S i would be stationary subsets of S * (id p (C,Ī) and thus they would reflect simultaneously. We now obtain Pr 1 (µ + , µ + , θ, cf(µ)) by way of Corollary 8.5.
As far as part (2) of our theorem goes, note that Theorem 3 gives us the squarebrackets part of the conclusion already. We will prove the remainder by contrapositive, focusing only on the case where µ is of countable cofinality because part (3) will take care of the other case.
An elementary argument by contradiction establishes the existence of a regular θ < µ (without loss of generality greater than ℵ 1 ) and an id p (C,Ī)-positive set A such that the ideal I = {B ⊆ µ + : A ∩ B ∈ id p (C,Ī)} is weakly θ-saturated. This ideal I is also τ -indecomposable for any regular τ with ℵ 1 < τ < µ -it inherits this property from id p (C,Ī). Thus, S * (I) is equal to S µ + ≥θ modulo the non-stationary ideal, and Refl(< cf(µ), S µ + ≥θ ) follows by Theorem 2. To finish, we note that parts (3) and (4) of our main theorem follow by exactly the same argument -for part (3) we take advantage of conclusion (4) of Theorem 8, and for part (4), we use Theorem 7 in the previous section.
We conclude our paper with a general discussion of some of the issues raised by the research presented here. First and foremost, it should be clear that we obtain simultaneous reflection almost by accident -it seems that a much more important phenomenon is isolated in Theorem 6 (and Theorem 2 of [7]), where we get in ZFC a coloring theorem intertwined with an easily describable ideal. In the presence of square-brackets partition relations, the ideal in question must possess some large cardinal type properties and it is not clear at this point if this is even possible: Question 2. Suppose µ is a singular cardinal. Is it consistent (relative to large cardinals) that ideals of the form id p (C,Ī) can be weakly θ-saturated for some θ ≤ µ?
If 2 µ = µ + , then an old theorem of Erdǒs, Hajnal, and Rado [?] tells us that µ + → [µ + ] 2 µ + holds. This motivates the following question, which we phrase in very specific terms: Question 3. Suppose µ is singular strong limit cardinal of countable cofinality. Is it consistent (relative to large cardinals) that 2 µ > µ + and there is a uniform ultrafilter U on µ + that is θ-indecomposable for all uncountable regular θ < µ?
The preceding question may be quite tractable -for example, if we ignore the cardinal arithmetic aspect of the question, then Ben-David and Magidor [2] show that such a filter can exist on ℵ ω . We are not sure of the extent to which the need for 2 µ to be greater than µ + complicates things.
Another somewhat bothersome point is the difference between parts (3) and (4) of our main theorem. This discrepancy may be resolvable by an easy argument, but as of yet we do not see how to do it. The general question is as follows: Question 4. Suppose µ is a singular cardinal for which Pr 1 (µ + , µ + , µ, cf(µ)) holds. Does Pr 1 (µ + , µ + , µ + , cf(µ)) hold?
Finally, having established that square-brackets-type relations at successors of singular cardinals necessarily entail simultaneous reflection of stationary sets, we would like to have information about the cofinalities of the ordinals where the reflection takes place: µ + for µ singular, and let θ < µ be a regular cardinal for which Refl(< cf(µ), S µ + ≥θ ) holds. Can we say anything interesting about the cofinalities of ordinals where the simultaneous reflection takes place?
The answer to the above depends on getting information about the cofinalities of ordinals in the range of f * , where f * is as in Theorem 2.