A NOTE ON STRONG NEGATIVE PARTITION RELATIONS
arXiv:math/0703647v2 [math.LO] 2 Jun 2008
A NOTE ON STRONG NEGATIVE PARTITION RELATIONS TODD EISWORTH Abstract. We analyze a natural function definable from a scale at a singular cardinal, and use it to obtain some strong negative square-brackets partition relations at successors of singular cardinals. The proof of our main result makes use of club-guessing, and as a corollary we obtain a fairly easy proof of a difficult result of Shelah connecting weak saturation of a certain clubguessing ideal with strong failures of square-brackets partition relations. We then investigate the strength of weak saturation of such ideals and obtain some results on stationary reflection.
1. Introduction Recall that the square-brackets partition relation κ → [λ]µθ of Erd˝ os, Hajnal, and Rado [7] asserts that for every function F : [κ]µ → θ (where [κ]µ denotes the subsets of κ of cardinality µ), there is a set H ⊆ κ of cardinality λ such that (1.1)
ran(F ↾ [H]µ ) 6= θ,
that is, the function F omits at least one value when we restrict it to [H]µ . The negations of square-brackets partition relations are particularly interesting, as such combinatorial principles have many applications outside of pure set theory. This paper is primarily concerned with the combinatorial statement (1.2)
λ 9 [λ]2λ ,
for λ the successor of a singular cardinal. The assertion (1.2) states that there exists a function F : [λ]2 → λ with the property that (1.3)
ran(F ↾ [A]2 ) = λ
for any unbounded subset A of λ. Traditionally, more descriptive language is used when discussing (1.2) — F is called a coloring, and (1.2) says that we can color the pairs of ordinals from λ using λ colors in such a way that all colors appear in any unbounded subset of λ. Thus, Ramsey’s Theorem fails for λ in an incredibly spectacular way. The question of whether λ 9 [λ]2λ necessarily holds for λ the successor of singular cardinal is still open. Much research has been devoted to this question (particularly by Shelah [10], [11], [13], [4]) and the related question of whether such a λ can be a J´onsson cardinal. This has resulted in a complex web of conditions that tightly constrains what a potential counterexample can look like, but still no proof that a counterexample cannot exist has emerged. Date: June 3, 2008. 1991 Mathematics Subject Classification. 03E02. Key words and phrases. successors of singular cardinals, scales, square brackets partition relations, club guessing. The author acknowledges support from NSF grant DMS 0506063. 1
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In this paper, we show (assuming λ = µ+ for µ singular) that in many situations there is a natural coloring c : [λ]2 → λ with the property that c takes on almost every color on every unbounded A ⊆ λ, We have used two qualifying phrases in the previous sentence. The first – “in many situations” – we leave vague for now, although our theorem is general enough to cover the case where µ is singular of uncountable cofinality. The second qualifying phrase – “almost every color” – means “the set of omitted colors is small in the sense that it lies in a certain ideal associated with Shelah’s theory of guessing clubs”. The proof that the coloring has the required characteristics is a blending of techniques due to Todorˇcevi´c (namely, the method of minimal walks [15, 16, 14]) and techniques due to Shelah (combinatorics associated with scales [10, 6]). We will actually we working with negative square-brackets partition relations much stronger than those discussed in the first two paragraphs of the paper. In particular, we will be investigating instances of the following, which is a specific case of a much more general property introduced studied by Shelah in several works [9, 13, 4, 12]. Definition 1. Let λ = µ+ for µ a singular cardinal, and suppose σ ≤ λ. We say Pr1 (λ, λ, σ, cf(µ)) holds if there is a function f : [λ]2 → σ such that whenever htα : α < λi is a sequence of pairwise disjoint elements of [λ] ǫ and cf(γ) > τ . ¯ I) ¯ is a proper ideal whenever (C, ¯ I) ¯ is an S-good pair because The ideal idp (C, of condition (5) in Definition 5. In this situation, the equation (2.3) means that for every τ < µ and ǫ < δ, there is a γ ∈ nacc(Cδ ) ∩ E ∩ A such that γ > ǫ and cf(γ) > τ . Minimal Walks Suppose now that e¯ = heα : α < λi is a C-sequence for some cardinal λ. Following Todorˇcevi´c, given α < β < λ the minimal walk from β to α along e¯ is defined to be the sequence β = β0 > · · · > βn+1 = α obtained by setting (2.7)
βi+1 = min(eβi \ α).
The use of “n + 1” as the index of the last step is deliberate, as the ordinal βn (the penultimate step) is quite important in our proof. The trace of the walk from β to α is defined by (2.8)
Tr(α, β) = {β = β0 > β1 > · · · > βn > βn+1 = α}.
We make use of standard facts about minimal walks. In particular, suppose δ is a limit ordinal, δ < β < λ, and β = β0 > · · · > βn+1 = δ is the minimal walk from β to δ. For i < n, we know that α ∈ / eβi , and so (2.9)
γ ∗ := max{max(eβi ∩ δ) : i < n} < δ.
Suppose now that γ ∗ < α < δ, and let β = β0∗ > · · · > βn∗ ∗ +1 = α be the minimal walk from β to α. From the definition of γ ∗ it follows that (2.10)
βi = βi∗ for i ≤ n.
Thus, the walks from β to δ and from β to α agree up to and and including the step before the former reaches its destination. A proper discussion of minimal walks and their applications is beyond the scope of this paper, and we develop the theory only to the degree that we need it for our proof. We refer the reader to [15], [2],[14], or [16] for more information.
A NOTE ON STRONG NEGATIVE PARTITION RELATIONS
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Scales The next ingredient we need for our theorem is the concept of a scale for a singular cardinal. Definition 6. Let µ be a singular cardinal. A scale for µ is a pair (~µ, f~) satisfying (1) µ ~ = hµi : i < cf(µ)i is an increasing sequence of regular cardinals such that supi δ. By Lemma 15, we know that Iδ -almost every member of Cδ is {β}-ok. If β ∗ ∈ E∩Cδ is {β}-ok, then for all sufficiently large α ∈ A∩β ∗ we have β ∗ ∈ Tr(α, β) and therefore ΓA contains Iδ -almost all members of E ∩ Cδ . 4. The main theorem We move now to the main theorem of this paper. Throughout this section, we adopt the following list of assumptions: Assumptions • λ = µ+ for µ singular of cofinality κ ¯ I) ¯ is an S-good pair for some stationary S ⊆ {δ < λ : cf(δ) = κ} • (C, ¯ I) ¯ • e¯ = heα : α < λi is a C-sequence that swallows (C, • (~ µ, f~) is a scale for µ • A = hH(χ), ∈, β1 > · · · > βn > βn+1 = α list Tr(α, β) in decreasing order. The function c : [λ]2 → λ is defined by setting c(α, β) equal to βm , where m ≤ n + 1 is the least number for which (4.1)
Γ(α, βm ) 6= Γ(α, β).
The function c can easily be described in English: to calculate the value of c(α, β), we first compute Γ(α, β), and then walk along e¯ until we reach an ordinal βm where Γ(α, βm ) is different from Γ(α, β). This ordinal βm is the value of c(α, β). Theorem 2. Suppose htα : α < λi is a sequence of pairwise disjoint subsets of λ, ¯ I)-almost ¯ each of cardinality less than κ. Then for idp (C, all β ∗ < λ the following holds: (∃∗ β < λ)(∀∗ i < κ)(∃∗ α < β ∗ )(∀ζ ∈ tα )(∀ξ ∈ tβ ) [c(ζ, ξ) = β ∗ ∧ Γ(ζ, ξ) = i] . Proof. We first prove the theorem under the assumptions that α < min(tα ) and sup(tα ) < min(tβ ) whenever α < β < λ. Given this, let A be the set of all β ∗ < λ for which the conclusion of the theorem fails, and assume by way of contradiction ¯ I)-positive. ¯ that A is idp (C, Let M = hMi : i < λi be a λ-approximating sequence over all the objects mentioned so far. We define E to be the set of δ < λ for which Mδ ∩ λ = δ. Since E
A NOTE ON STRONG NEGATIVE PARTITION RELATIONS
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¯ I), ¯ we know there is a δ ∈ S ∩ E is a closed unbounded subset of λ and A ∈ / idp (C, with A ∩ E ∩ Cδ ∈ / Iδ .
(4.2)
We have assumed δ < min(tδ ), so for each ξ ∈ tδ let ξ ξ ξ = β0ξ > · · · > βn(ξ) > βn(ξ)+1 =δ
(4.3)
list Tr(δ, ξ) in decreasing order. Taking (4.2) together with Lemma 15, we can find β ∗ ∈ A ∩ E ∩ Cδ and γ ∗ < β ∗ with cf(β ∗ ) > κ so that whenever γ ∗ < α < β ∗ and ξ ∈ tδ the walk along e¯ from ξ to α commences with the sequence ξ ξ = β0ξ > · · · > βn(ξ) > β∗.
(4.4)
We will now prove that the following statement holds: (4.5)
(∀∗ i < κ)(∃∗ α < β ∗ )(∀ζ ∈ tα )(∀ξ ∈ tδ ) [c(ζ, ξ) = β ∗ ∧ Γ(ζ, ξ) = i] .
If we combine the above with the fact that sup(Mβ ∗ +1 ∩ λ) ∈ Mδ ∩ λ = δ, we find that Mβ ∗ +1 |= β ∗ ∈ / A, which is a contradiction. For each α < λ, we define a function fαmin as follows: fαmin (i) = min{fζ (i) : ζ ∈ tα }.
(4.6)
Since |tα | < κ for each α, it follows that (4.7) (∀∗ i < κ) fαmin (i) = fmin(tα ) (i) .
In particular, the sequence hfαmin : α < λi is a scale, and we note that this new scale is an element of M0 . An appeal to Lemma 7 gives us a closed unbounded set in M0 as there. In particular, this closed unbounded set is also an element of Mβ ∗ and so β ∗ = sup(Mβ ∗ ∩ λ) is necessarily a member of this closed unbounded set. We conclude (∀∗ i < κ)(∀η < µi )(∀ν < µi+1 )(∃∗ α < β ∗ ) fαmin(i) > η ∧ fαmin(i + 1) > ν . This means we can choose i0 < κ such that
i0 ≤ i < κ =⇒ (∀η < µi )(∀ν < µi+1 )(∃∗ α < β ∗ ) fαmin (i) > η ∧ fαmin (i + 1) > ν . The next piece of the proof makes use of Skolem hulls. Let us define x := {λ, µ, κ, (~µ, f~), S, e¯, htα : α < λi, β ∗ }, and define M to be the Skolem hull in A of x together with all ordinals less than or equal to κ, that is, M := SkA (x ∪ κ + 1). Since |M | = κ < µ0 , it follows that ChM (i) = sup(M ∩ µi ) for all i < κ,
(4.8)
where ChM is the characteristic function of M from Definition 10. We note that M can be computed by taking a Skolem hull in the model Mβ ∗ +1 and therefore M ∈ Mδ . In particular, Y µi (4.9) ChM ∈ Mδ ∩ i η ∗ ∧ fαmin(i + 1) > ν ∗ . min(i)
Suppose now that α ∈ N satisfies γ ∗ < α < β, fα From (4.15) and the choice of η ∗ , we conclude
(4.17)
> η ∗ , and fαmin (i+1) > ν ∗ .
Γ(ζ, βjξ ) = i for all ζ ∈ tα , ξ ∈ tδ , and j ≤ n(ξ).
By our choice of ν ∗ , we know (4.18)
Γ(ζ, β ∗ ) ≥ i + 1 for all ζ ∈ tα .
The conjunction of (4.17) and (4.18) establishes (4.19)
(∀ζ ∈ tα )(∀ξ ∈ tδ ) [c(ζ, ξ) = β ∗ ∧ Γ(ζ, ξ) = i] .
Thus, (4.5) holds and a contradiction arises because β ∗ was chosen to be in the set A. We now fulfill the promise made at the start of the proof by handling the case of an arbitrary sequence htα : α < λi. Given such a sequence, we define an increasing function ι : λ → λ such that • α < min(tι(α) ) for all α < λ, and • α < β < λ =⇒ max(tα ) < ι(β).
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If we define sα = tι(α) , then our work applies to the sequence hsα : α < λi. In ¯ I) ¯ so that the conclusion of particular, there is a set B in the filter dual to idp (C, our theorem (as it applies to hsα : α < λi) holds for every β ∗ ∈ B. Let B ∗ be the set of all β ∗ ∈ B that are closed under the function ι. Since ¯ I) ¯ extends the non-stationary ideal, it is clear that B ∗ is in the filter dual to idp (C, ¯ I) ¯ and routine to check that the conclusion of the theorem (as it applies to idp (C, htα : α < λi) holds for all β ∗ ∈ B ∗ . We will shortly deduce an interesting theorem of Shelah as a corollary to our main result, but to do this we need to fix some terminology. Definition 18. Let I be an ideal on some set A, and let σ be a cardinal. The ideal I is weakly σ-saturated if A cannot be partitioned into σ disjoint I-positive sets, i.e., there is no function π : A → σ such that π −1 (i) ∈ /I for all i < σ. It is clear that any maximal ideal is weakly 2-saturated, so weakly saturated ideals are not very difficult to find. The rest of this paper will demonstrate that ¯ I)?” ¯ the question of “how weakly saturated is idp (C, is quite important. We begin with the following, which follows from the work in Section 4 of Shelah’s [11]. ¯ I) ¯ is an S-good pair for Corollary 19. Suppose λ = µ+ for µ singular, and (C, ¯ I) ¯ is not weakly some stationary subset S of {δ < λ : cf(δ) = cf(µ)}. If idp (C, σ-saturated, then Pr1 (λ, λ, σ, cf(µ)) holds; in particular, we have λ 9 [σ]2λ . ¯ I)-positive ¯ Proof. Let π : λ → σ partition λ into disjoint idp (C, sets, and define 2 f : [λ] → σ by f (α, β) = π(c(α, β)). Suppose htα : α < λi is a family of disjoint subsets of λ each of cardinality less than cf(µ), and let ǫ < σ be given. By Theorem 2, we can find α < β ∗ < β such that π(β ∗ ) = ǫ and c(ζ, ξ) = β ∗ for all ζ ∈ tα and ξ ∈ tβ . It is clear that f is constant with value ǫ when restricted to tα × tβ . Shelah’s original proof of the above is much more difficult, as he starts with a ¯ I)-positive ¯ partition of λ into idp (C, sets and uses this as a parameter to define his coloring, whereas we use a scale to get a single “master coloring” that can be used (in the sense of the proof of Corollary 19) in conjunction with any such partition. 5. From µ to µ+ ¯ I) ¯ is an S-good pair for some Let µ be a singular cardinal, and suppose (C, + stationary S ⊆ {δ < µ : cf(δ) = cf(µ)}. The results of the previous section ¯ I). ¯ In focused our attention on the degree of weak saturation possessed by idp (C, this section we get an improvement of Corollary 19. ¯ I) ¯ is not weakly µ-saturated, then Corollary 19 tells us If we assume that idp (C, (5.1)
Pr1 (µ+ , µ+ , µ, cf(µ))
holds, and we immediately obtain the relation (5.2)
µ+ 9 [µ]2µ+ .
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Now an elementary argument allows us to “upgrade” the relation (5.2) to the case of µ+ colors, that is, we can easily obtain the stronger result µ+ 9 [µ+ ]2µ+
(5.3)
from (5.2). It is natural to ask if we can also upgrade (5.1) to obtain Pr1 (µ+ , µ+ , µ+ , cf(µ)).
(5.4)
We do not know if (5.4) follows from (5.1) in general, but we have as consolation the following new theorem that tells us that (5.4) can be obtained from the same hypotheses we use to obtain (5.1). ¯ I) ¯ is an STheorem 3. Suppose λ = µ+ for µ a singular cardinal, and suppose (C, good pair for some stationary S ⊆ {δ < λ : cf(δ) = cf(µ)}. Then Pr1 (µ+ , µ+ , µ+ , cf(µ)) ¯ I) ¯ is not weakly µ-saturated. holds if idp (C, Proof. For each α < λ we fix surjection gα from µ onto α, and let π : λ → µ ¯ I)-positive ¯ give a partition of λ into disjoint idp (C, sets. We also fix a function −1 h : cf(µ) → ω such that h ({n}) is unbounded in cf(µ) for each n < ω, and let e¯ ¯ I). ¯ be a C-sequence swallowing (C, Given α < β, let β = β0 > · · · > βn > βn+1 = α list Tr(α, β) (where we walk along e¯, just as in the proof of Theorem 2) in decreasing order, and let i∗ denote Γ(α, β). We define m(α, β) to be the least m ≤ n + 1 with Γ(α, βm ) 6= i∗ ,
(5.5)
so that in terms of the coloring from Section 4, we have (5.6)
c(α, β) = βm(α,β) .
We also define (5.7)
k(α, β) =
(
m(α, β) − h(Γ(α, β)) 0
if h(Γ(α, β)) ≤ m(α, β), otherwise
Finally, we define (5.8)
c∗ (α, β) = gβk(α,β) π ◦ c(α, β) .
A formula like (5.8) surely deserves some explanation, so we will describe the coloring we use in English. Given α < β, we first compute i∗ = Γ(α, β) and note that h(i∗ ) is some natural number. We then walk from β to α until the first place where Γ changes. This isolates βm(α,β) = c(α, β), and π(c(α, β)) records the piece of the partition that contains the ordinal βm(α,β) . Next, we turn around and retrace h(i∗ ) steps of the walk from β to βm(α,β) (so we are walking up, not down). This takes us to an ordinal (5.9)
βk(α,β) > βm(α,β) .
Now to compute the value of c (α, β), we take the bijection between µ and βk(α,β) and apply it to the ordinal π(βm(α,β) ). We now prove that the coloring c∗ has the properties required by Pr1 (λ, λ, λ, cf(µ)). Let hti : i < λi be a sequence of pairwise disjoint elements of [λ] β1 > · · · > βn = ζ list Tr(ζ, ξ) in decreasing order. Since c(ζ, ξ) = β ∗ , we know that β ∗ = βm(ζ,ξ) . Now γ ∗ < ζ < β ∗ < β < ξ, and so the choice of γ ∗ implies that the walk from ξ to ζ must pass through β before proceeding on to β ∗ . Since h(Γ(ζ, ξ)) is the length of the walk from β to β ∗ , we know βk(ζ,ξ) = β. Since gβ (ς ∗ ) = ǫ and π(β ∗ ) = ς ∗ , we conclude c∗ (ζ, ξ) = ǫ, as required.
In the next section, we show that in our usual context, if an ideal of the form ¯ I) ¯ is weakly µ-saturated for µ strong limit singular then every stationary idp (C, subset of {δ < µ+ : cf(µ) 6= cf(δ)} reflects.
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¯ I) ¯ 6. On the weak saturation of idp (C, In this section, we directly address the question of weak saturation of ideals of ¯ I). ¯ Our results do not require the full strength of Definition 5. the form idp (C, In particular, the requirement (2), although important for the arguments in the preceding two theorems, is not a necessary ingredient in the proofs of this section. With this in mind, we offer the following definition in the same spirit as Definition 5. Definition 20. Suppose λ = µ+ for µ a singular cardinal, and S is a stationary ¯ I) ¯ is an S-fair pair if the subset of {δ < λ : cf(δ) = cf(µ)}. We say that (C, following conditions are satisfied: (1) C¯ = hCδ : δ ∈ Si is an S-club sequence (2) I¯ = hIδ : δ ∈ Si (3) for δ ∈ S, Iδ is the ideal on Cδ generated by sets of the form (6.1)
{γ ∈ Cδ : γ ∈ acc(Cδ ) or cf(γ) < α or γ < β}
for α < µ and β < δ. (4) for every closed unbounded E ⊆ λ, we have (6.2)
{δ ∈ S : E ∩ Cδ ∈ / Iδ } is stationary.
It is clear that any S-good pair is also S-fair. In contrast to S-good pairs, it is known that S-fair pairs exist for any stationary S ⊆ {δ < µ+ : cf(µ) = cf(δ)} regardless of the cofinality of µ — the countable cofinality case is handled by Claim 2.8 of page 131 in [9], while another proof (yielding more information) can be found in [5]. ¯ I) ¯ associated with Definition 3 still applies, so we have a proper ideal idp (C, every S-fair pair. We note the following facts about this ideal: ¯ I) ¯ is an S-fair pair Proposition 21. Let λ = µ+ for µ singular, and suppose (C, for some stationary subset S of {δ < λ : cf(δ) = cf(µ)}. ¯ I) ¯ is closed under unions of fewer than cf(µ) sets. (1) The ideal idp (C, ¯ I) ¯ such (2) There is an increasing sequence hAi : i < cf(µ)i of sets in idp (C, that [ Ai . λ= i
A NOTE ON STRONG NEGATIVE PARTITION RELATIONS TODD EISWORTH Abstract. We analyze a natural function definable from a scale at a singular cardinal, and use it to obtain some strong negative square-brackets partition relations at successors of singular cardinals. The proof of our main result makes use of club-guessing, and as a corollary we obtain a fairly easy proof of a difficult result of Shelah connecting weak saturation of a certain clubguessing ideal with strong failures of square-brackets partition relations. We then investigate the strength of weak saturation of such ideals and obtain some results on stationary reflection.
1. Introduction Recall that the square-brackets partition relation κ → [λ]µθ of Erd˝ os, Hajnal, and Rado [7] asserts that for every function F : [κ]µ → θ (where [κ]µ denotes the subsets of κ of cardinality µ), there is a set H ⊆ κ of cardinality λ such that (1.1)
ran(F ↾ [H]µ ) 6= θ,
that is, the function F omits at least one value when we restrict it to [H]µ . The negations of square-brackets partition relations are particularly interesting, as such combinatorial principles have many applications outside of pure set theory. This paper is primarily concerned with the combinatorial statement (1.2)
λ 9 [λ]2λ ,
for λ the successor of a singular cardinal. The assertion (1.2) states that there exists a function F : [λ]2 → λ with the property that (1.3)
ran(F ↾ [A]2 ) = λ
for any unbounded subset A of λ. Traditionally, more descriptive language is used when discussing (1.2) — F is called a coloring, and (1.2) says that we can color the pairs of ordinals from λ using λ colors in such a way that all colors appear in any unbounded subset of λ. Thus, Ramsey’s Theorem fails for λ in an incredibly spectacular way. The question of whether λ 9 [λ]2λ necessarily holds for λ the successor of singular cardinal is still open. Much research has been devoted to this question (particularly by Shelah [10], [11], [13], [4]) and the related question of whether such a λ can be a J´onsson cardinal. This has resulted in a complex web of conditions that tightly constrains what a potential counterexample can look like, but still no proof that a counterexample cannot exist has emerged. Date: June 3, 2008. 1991 Mathematics Subject Classification. 03E02. Key words and phrases. successors of singular cardinals, scales, square brackets partition relations, club guessing. The author acknowledges support from NSF grant DMS 0506063. 1
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TODD EISWORTH
In this paper, we show (assuming λ = µ+ for µ singular) that in many situations there is a natural coloring c : [λ]2 → λ with the property that c takes on almost every color on every unbounded A ⊆ λ, We have used two qualifying phrases in the previous sentence. The first – “in many situations” – we leave vague for now, although our theorem is general enough to cover the case where µ is singular of uncountable cofinality. The second qualifying phrase – “almost every color” – means “the set of omitted colors is small in the sense that it lies in a certain ideal associated with Shelah’s theory of guessing clubs”. The proof that the coloring has the required characteristics is a blending of techniques due to Todorˇcevi´c (namely, the method of minimal walks [15, 16, 14]) and techniques due to Shelah (combinatorics associated with scales [10, 6]). We will actually we working with negative square-brackets partition relations much stronger than those discussed in the first two paragraphs of the paper. In particular, we will be investigating instances of the following, which is a specific case of a much more general property introduced studied by Shelah in several works [9, 13, 4, 12]. Definition 1. Let λ = µ+ for µ a singular cardinal, and suppose σ ≤ λ. We say Pr1 (λ, λ, σ, cf(µ)) holds if there is a function f : [λ]2 → σ such that whenever htα : α < λi is a sequence of pairwise disjoint elements of [λ] ǫ and cf(γ) > τ . ¯ I) ¯ is a proper ideal whenever (C, ¯ I) ¯ is an S-good pair because The ideal idp (C, of condition (5) in Definition 5. In this situation, the equation (2.3) means that for every τ < µ and ǫ < δ, there is a γ ∈ nacc(Cδ ) ∩ E ∩ A such that γ > ǫ and cf(γ) > τ . Minimal Walks Suppose now that e¯ = heα : α < λi is a C-sequence for some cardinal λ. Following Todorˇcevi´c, given α < β < λ the minimal walk from β to α along e¯ is defined to be the sequence β = β0 > · · · > βn+1 = α obtained by setting (2.7)
βi+1 = min(eβi \ α).
The use of “n + 1” as the index of the last step is deliberate, as the ordinal βn (the penultimate step) is quite important in our proof. The trace of the walk from β to α is defined by (2.8)
Tr(α, β) = {β = β0 > β1 > · · · > βn > βn+1 = α}.
We make use of standard facts about minimal walks. In particular, suppose δ is a limit ordinal, δ < β < λ, and β = β0 > · · · > βn+1 = δ is the minimal walk from β to δ. For i < n, we know that α ∈ / eβi , and so (2.9)
γ ∗ := max{max(eβi ∩ δ) : i < n} < δ.
Suppose now that γ ∗ < α < δ, and let β = β0∗ > · · · > βn∗ ∗ +1 = α be the minimal walk from β to α. From the definition of γ ∗ it follows that (2.10)
βi = βi∗ for i ≤ n.
Thus, the walks from β to δ and from β to α agree up to and and including the step before the former reaches its destination. A proper discussion of minimal walks and their applications is beyond the scope of this paper, and we develop the theory only to the degree that we need it for our proof. We refer the reader to [15], [2],[14], or [16] for more information.
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5
Scales The next ingredient we need for our theorem is the concept of a scale for a singular cardinal. Definition 6. Let µ be a singular cardinal. A scale for µ is a pair (~µ, f~) satisfying (1) µ ~ = hµi : i < cf(µ)i is an increasing sequence of regular cardinals such that supi δ. By Lemma 15, we know that Iδ -almost every member of Cδ is {β}-ok. If β ∗ ∈ E∩Cδ is {β}-ok, then for all sufficiently large α ∈ A∩β ∗ we have β ∗ ∈ Tr(α, β) and therefore ΓA contains Iδ -almost all members of E ∩ Cδ . 4. The main theorem We move now to the main theorem of this paper. Throughout this section, we adopt the following list of assumptions: Assumptions • λ = µ+ for µ singular of cofinality κ ¯ I) ¯ is an S-good pair for some stationary S ⊆ {δ < λ : cf(δ) = κ} • (C, ¯ I) ¯ • e¯ = heα : α < λi is a C-sequence that swallows (C, • (~ µ, f~) is a scale for µ • A = hH(χ), ∈, β1 > · · · > βn > βn+1 = α list Tr(α, β) in decreasing order. The function c : [λ]2 → λ is defined by setting c(α, β) equal to βm , where m ≤ n + 1 is the least number for which (4.1)
Γ(α, βm ) 6= Γ(α, β).
The function c can easily be described in English: to calculate the value of c(α, β), we first compute Γ(α, β), and then walk along e¯ until we reach an ordinal βm where Γ(α, βm ) is different from Γ(α, β). This ordinal βm is the value of c(α, β). Theorem 2. Suppose htα : α < λi is a sequence of pairwise disjoint subsets of λ, ¯ I)-almost ¯ each of cardinality less than κ. Then for idp (C, all β ∗ < λ the following holds: (∃∗ β < λ)(∀∗ i < κ)(∃∗ α < β ∗ )(∀ζ ∈ tα )(∀ξ ∈ tβ ) [c(ζ, ξ) = β ∗ ∧ Γ(ζ, ξ) = i] . Proof. We first prove the theorem under the assumptions that α < min(tα ) and sup(tα ) < min(tβ ) whenever α < β < λ. Given this, let A be the set of all β ∗ < λ for which the conclusion of the theorem fails, and assume by way of contradiction ¯ I)-positive. ¯ that A is idp (C, Let M = hMi : i < λi be a λ-approximating sequence over all the objects mentioned so far. We define E to be the set of δ < λ for which Mδ ∩ λ = δ. Since E
A NOTE ON STRONG NEGATIVE PARTITION RELATIONS
11
¯ I), ¯ we know there is a δ ∈ S ∩ E is a closed unbounded subset of λ and A ∈ / idp (C, with A ∩ E ∩ Cδ ∈ / Iδ .
(4.2)
We have assumed δ < min(tδ ), so for each ξ ∈ tδ let ξ ξ ξ = β0ξ > · · · > βn(ξ) > βn(ξ)+1 =δ
(4.3)
list Tr(δ, ξ) in decreasing order. Taking (4.2) together with Lemma 15, we can find β ∗ ∈ A ∩ E ∩ Cδ and γ ∗ < β ∗ with cf(β ∗ ) > κ so that whenever γ ∗ < α < β ∗ and ξ ∈ tδ the walk along e¯ from ξ to α commences with the sequence ξ ξ = β0ξ > · · · > βn(ξ) > β∗.
(4.4)
We will now prove that the following statement holds: (4.5)
(∀∗ i < κ)(∃∗ α < β ∗ )(∀ζ ∈ tα )(∀ξ ∈ tδ ) [c(ζ, ξ) = β ∗ ∧ Γ(ζ, ξ) = i] .
If we combine the above with the fact that sup(Mβ ∗ +1 ∩ λ) ∈ Mδ ∩ λ = δ, we find that Mβ ∗ +1 |= β ∗ ∈ / A, which is a contradiction. For each α < λ, we define a function fαmin as follows: fαmin (i) = min{fζ (i) : ζ ∈ tα }.
(4.6)
Since |tα | < κ for each α, it follows that (4.7) (∀∗ i < κ) fαmin (i) = fmin(tα ) (i) .
In particular, the sequence hfαmin : α < λi is a scale, and we note that this new scale is an element of M0 . An appeal to Lemma 7 gives us a closed unbounded set in M0 as there. In particular, this closed unbounded set is also an element of Mβ ∗ and so β ∗ = sup(Mβ ∗ ∩ λ) is necessarily a member of this closed unbounded set. We conclude (∀∗ i < κ)(∀η < µi )(∀ν < µi+1 )(∃∗ α < β ∗ ) fαmin(i) > η ∧ fαmin(i + 1) > ν . This means we can choose i0 < κ such that
i0 ≤ i < κ =⇒ (∀η < µi )(∀ν < µi+1 )(∃∗ α < β ∗ ) fαmin (i) > η ∧ fαmin (i + 1) > ν . The next piece of the proof makes use of Skolem hulls. Let us define x := {λ, µ, κ, (~µ, f~), S, e¯, htα : α < λi, β ∗ }, and define M to be the Skolem hull in A of x together with all ordinals less than or equal to κ, that is, M := SkA (x ∪ κ + 1). Since |M | = κ < µ0 , it follows that ChM (i) = sup(M ∩ µi ) for all i < κ,
(4.8)
where ChM is the characteristic function of M from Definition 10. We note that M can be computed by taking a Skolem hull in the model Mβ ∗ +1 and therefore M ∈ Mδ . In particular, Y µi (4.9) ChM ∈ Mδ ∩ i η ∗ ∧ fαmin(i + 1) > ν ∗ . min(i)
Suppose now that α ∈ N satisfies γ ∗ < α < β, fα From (4.15) and the choice of η ∗ , we conclude
(4.17)
> η ∗ , and fαmin (i+1) > ν ∗ .
Γ(ζ, βjξ ) = i for all ζ ∈ tα , ξ ∈ tδ , and j ≤ n(ξ).
By our choice of ν ∗ , we know (4.18)
Γ(ζ, β ∗ ) ≥ i + 1 for all ζ ∈ tα .
The conjunction of (4.17) and (4.18) establishes (4.19)
(∀ζ ∈ tα )(∀ξ ∈ tδ ) [c(ζ, ξ) = β ∗ ∧ Γ(ζ, ξ) = i] .
Thus, (4.5) holds and a contradiction arises because β ∗ was chosen to be in the set A. We now fulfill the promise made at the start of the proof by handling the case of an arbitrary sequence htα : α < λi. Given such a sequence, we define an increasing function ι : λ → λ such that • α < min(tι(α) ) for all α < λ, and • α < β < λ =⇒ max(tα ) < ι(β).
A NOTE ON STRONG NEGATIVE PARTITION RELATIONS
13
If we define sα = tι(α) , then our work applies to the sequence hsα : α < λi. In ¯ I) ¯ so that the conclusion of particular, there is a set B in the filter dual to idp (C, our theorem (as it applies to hsα : α < λi) holds for every β ∗ ∈ B. Let B ∗ be the set of all β ∗ ∈ B that are closed under the function ι. Since ¯ I) ¯ extends the non-stationary ideal, it is clear that B ∗ is in the filter dual to idp (C, ¯ I) ¯ and routine to check that the conclusion of the theorem (as it applies to idp (C, htα : α < λi) holds for all β ∗ ∈ B ∗ . We will shortly deduce an interesting theorem of Shelah as a corollary to our main result, but to do this we need to fix some terminology. Definition 18. Let I be an ideal on some set A, and let σ be a cardinal. The ideal I is weakly σ-saturated if A cannot be partitioned into σ disjoint I-positive sets, i.e., there is no function π : A → σ such that π −1 (i) ∈ /I for all i < σ. It is clear that any maximal ideal is weakly 2-saturated, so weakly saturated ideals are not very difficult to find. The rest of this paper will demonstrate that ¯ I)?” ¯ the question of “how weakly saturated is idp (C, is quite important. We begin with the following, which follows from the work in Section 4 of Shelah’s [11]. ¯ I) ¯ is an S-good pair for Corollary 19. Suppose λ = µ+ for µ singular, and (C, ¯ I) ¯ is not weakly some stationary subset S of {δ < λ : cf(δ) = cf(µ)}. If idp (C, σ-saturated, then Pr1 (λ, λ, σ, cf(µ)) holds; in particular, we have λ 9 [σ]2λ . ¯ I)-positive ¯ Proof. Let π : λ → σ partition λ into disjoint idp (C, sets, and define 2 f : [λ] → σ by f (α, β) = π(c(α, β)). Suppose htα : α < λi is a family of disjoint subsets of λ each of cardinality less than cf(µ), and let ǫ < σ be given. By Theorem 2, we can find α < β ∗ < β such that π(β ∗ ) = ǫ and c(ζ, ξ) = β ∗ for all ζ ∈ tα and ξ ∈ tβ . It is clear that f is constant with value ǫ when restricted to tα × tβ . Shelah’s original proof of the above is much more difficult, as he starts with a ¯ I)-positive ¯ partition of λ into idp (C, sets and uses this as a parameter to define his coloring, whereas we use a scale to get a single “master coloring” that can be used (in the sense of the proof of Corollary 19) in conjunction with any such partition. 5. From µ to µ+ ¯ I) ¯ is an S-good pair for some Let µ be a singular cardinal, and suppose (C, + stationary S ⊆ {δ < µ : cf(δ) = cf(µ)}. The results of the previous section ¯ I). ¯ In focused our attention on the degree of weak saturation possessed by idp (C, this section we get an improvement of Corollary 19. ¯ I) ¯ is not weakly µ-saturated, then Corollary 19 tells us If we assume that idp (C, (5.1)
Pr1 (µ+ , µ+ , µ, cf(µ))
holds, and we immediately obtain the relation (5.2)
µ+ 9 [µ]2µ+ .
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TODD EISWORTH
Now an elementary argument allows us to “upgrade” the relation (5.2) to the case of µ+ colors, that is, we can easily obtain the stronger result µ+ 9 [µ+ ]2µ+
(5.3)
from (5.2). It is natural to ask if we can also upgrade (5.1) to obtain Pr1 (µ+ , µ+ , µ+ , cf(µ)).
(5.4)
We do not know if (5.4) follows from (5.1) in general, but we have as consolation the following new theorem that tells us that (5.4) can be obtained from the same hypotheses we use to obtain (5.1). ¯ I) ¯ is an STheorem 3. Suppose λ = µ+ for µ a singular cardinal, and suppose (C, good pair for some stationary S ⊆ {δ < λ : cf(δ) = cf(µ)}. Then Pr1 (µ+ , µ+ , µ+ , cf(µ)) ¯ I) ¯ is not weakly µ-saturated. holds if idp (C, Proof. For each α < λ we fix surjection gα from µ onto α, and let π : λ → µ ¯ I)-positive ¯ give a partition of λ into disjoint idp (C, sets. We also fix a function −1 h : cf(µ) → ω such that h ({n}) is unbounded in cf(µ) for each n < ω, and let e¯ ¯ I). ¯ be a C-sequence swallowing (C, Given α < β, let β = β0 > · · · > βn > βn+1 = α list Tr(α, β) (where we walk along e¯, just as in the proof of Theorem 2) in decreasing order, and let i∗ denote Γ(α, β). We define m(α, β) to be the least m ≤ n + 1 with Γ(α, βm ) 6= i∗ ,
(5.5)
so that in terms of the coloring from Section 4, we have (5.6)
c(α, β) = βm(α,β) .
We also define (5.7)
k(α, β) =
(
m(α, β) − h(Γ(α, β)) 0
if h(Γ(α, β)) ≤ m(α, β), otherwise
Finally, we define (5.8)
c∗ (α, β) = gβk(α,β) π ◦ c(α, β) .
A formula like (5.8) surely deserves some explanation, so we will describe the coloring we use in English. Given α < β, we first compute i∗ = Γ(α, β) and note that h(i∗ ) is some natural number. We then walk from β to α until the first place where Γ changes. This isolates βm(α,β) = c(α, β), and π(c(α, β)) records the piece of the partition that contains the ordinal βm(α,β) . Next, we turn around and retrace h(i∗ ) steps of the walk from β to βm(α,β) (so we are walking up, not down). This takes us to an ordinal (5.9)
βk(α,β) > βm(α,β) .
Now to compute the value of c (α, β), we take the bijection between µ and βk(α,β) and apply it to the ordinal π(βm(α,β) ). We now prove that the coloring c∗ has the properties required by Pr1 (λ, λ, λ, cf(µ)). Let hti : i < λi be a sequence of pairwise disjoint elements of [λ] β1 > · · · > βn = ζ list Tr(ζ, ξ) in decreasing order. Since c(ζ, ξ) = β ∗ , we know that β ∗ = βm(ζ,ξ) . Now γ ∗ < ζ < β ∗ < β < ξ, and so the choice of γ ∗ implies that the walk from ξ to ζ must pass through β before proceeding on to β ∗ . Since h(Γ(ζ, ξ)) is the length of the walk from β to β ∗ , we know βk(ζ,ξ) = β. Since gβ (ς ∗ ) = ǫ and π(β ∗ ) = ς ∗ , we conclude c∗ (ζ, ξ) = ǫ, as required.
In the next section, we show that in our usual context, if an ideal of the form ¯ I) ¯ is weakly µ-saturated for µ strong limit singular then every stationary idp (C, subset of {δ < µ+ : cf(µ) 6= cf(δ)} reflects.
16
TODD EISWORTH
¯ I) ¯ 6. On the weak saturation of idp (C, In this section, we directly address the question of weak saturation of ideals of ¯ I). ¯ Our results do not require the full strength of Definition 5. the form idp (C, In particular, the requirement (2), although important for the arguments in the preceding two theorems, is not a necessary ingredient in the proofs of this section. With this in mind, we offer the following definition in the same spirit as Definition 5. Definition 20. Suppose λ = µ+ for µ a singular cardinal, and S is a stationary ¯ I) ¯ is an S-fair pair if the subset of {δ < λ : cf(δ) = cf(µ)}. We say that (C, following conditions are satisfied: (1) C¯ = hCδ : δ ∈ Si is an S-club sequence (2) I¯ = hIδ : δ ∈ Si (3) for δ ∈ S, Iδ is the ideal on Cδ generated by sets of the form (6.1)
{γ ∈ Cδ : γ ∈ acc(Cδ ) or cf(γ) < α or γ < β}
for α < µ and β < δ. (4) for every closed unbounded E ⊆ λ, we have (6.2)
{δ ∈ S : E ∩ Cδ ∈ / Iδ } is stationary.
It is clear that any S-good pair is also S-fair. In contrast to S-good pairs, it is known that S-fair pairs exist for any stationary S ⊆ {δ < µ+ : cf(µ) = cf(δ)} regardless of the cofinality of µ — the countable cofinality case is handled by Claim 2.8 of page 131 in [9], while another proof (yielding more information) can be found in [5]. ¯ I) ¯ associated with Definition 3 still applies, so we have a proper ideal idp (C, every S-fair pair. We note the following facts about this ideal: ¯ I) ¯ is an S-fair pair Proposition 21. Let λ = µ+ for µ singular, and suppose (C, for some stationary subset S of {δ < λ : cf(δ) = cf(µ)}. ¯ I) ¯ is closed under unions of fewer than cf(µ) sets. (1) The ideal idp (C, ¯ I) ¯ such (2) There is an increasing sequence hAi : i < cf(µ)i of sets in idp (C, that [ Ai . λ= i
