Section 1 of the Appendix to Shelah’s Proper and Improper Forcing (on weak diamond), retyped

with minor modifications

Paul Larson

March 16, 2010

0.1 Definition. Weak Diamond (Φ) is the statement that for every function F: ^{<ω_{1}}2 → 2 there exists an h: ω_{1} → 2 such that for all η: ω_{1} → 2 the set {α < ω_{1} | F (η↾α) = h(α)} is stationary.

Note that by replacing h with 1 − h, Φ is equivalent to statement that for every F: ^{<ω_{1}}2 → 2 there exists an h: ω_{1} → 2 such that for all η: ω_{1} → 2 the set {α < ω_{1} | F (η↾α) ≠ h(α)} is stationary.

Weak Diamond can be generalized to arbitrary cardinals as follows.

0.2 Definition. Given a cardinal κ and an ordinal λ of uncountable cofinality, Φ_{λ}^{κ} is the statement that for every F: ^{<λ}2 → κ there exists an h: λ → κ such that for all η: λ → 2 the set {α < λ | F (η↾α) = h(α)} is stationary.

Under this terminology, Φ is Φ_{ω_{1}}^{2}.

Notation. For any ordinal γ, for any α ≤ γ and any sequence of nonzero ordinals Ē = ⟨E(i)i < γ⟩ of length γ, let D_{α}(E) denote Π_{i<α}E(i), and let D(Ē) denote ⋃_{α<γ} Π_{i<α}E(i). When Ē has the constant value γ, we write D(γ) for D(Ē) and D_{α}(γ) for D_{α}(Ē).

The negation of Φ can be generalized as follows.

0.3 Definition. Suppose that λ is an ordinal of uncountable cofinality, and let μ̄ = ⟨μ̄(i) : i < λ⟩ and χ̄ = ⟨χ̄(i) : i < λ⟩ be sequences of nonzero cardinals. Unif(λ, μ̄, χ̄) is the statement that there is a function F with domain D(μ̄) such that F (η) ∈ χ̄(α) for each η ∈ D_{α}(μ̄) such that for every h ∈ D_{λ}(χ̄) there exists an η ∈ D_{λ}(μ̄) such that {α < λ | F (η↾α) = h(α)} contains a club subset of λ.

When μ̄ or χ̄ are constant, we write the constant value μ in place of μ̄ (and similarly for χ̄. So, for example, Unif(λ, 2, 2) is the negation of Φ_{λ}^{2}.

Notation. When λ is understood, ⟨μ(0), μ(1)⟩ denotes the sequence of length λ whose first element is μ(0) and whose other elements are all μ(1). We let D(μ(0), μ(1)) and D_{α}(μ(0), μ(1)) denote the corresponding versions of D(Ē) and D_{α}(Ē).

0.4 Definition. Given S ⊆ λ, Unif(λ, S, μ̄, χ̄) denotes the statement that there is a function F with domain D(μ̄) such that F (η) ∈ χ̄(α) for each α ∈ D_{α}(μ̄) such that for every h ∈ D_{λ}(χ̄) there exists an η ∈ D_{λ}(μ̄) such that

{α ∈ S | F (η↾α) = h(α)}

contains a relative club subset of S.

Again, we can replace μ̄ with μ_{0}, μ_{1}, so Unif can take up to five arguments and as few as three.

0.5 Definition. Id-Unif(λ, μ̄, χ̄) is the set of S ⊆ λ such that Unif(λ, S, μ̄, χ̄) holds.

Note that Id-Unif(λ, μ̄, χ̄) trivially contains all nonstationary subsets of λ. The following facts (Lemma 1.4) are straightforward.

Lemma 0.6. 1. If χ̄(i) = 1 for club many i, then Unif(λ, S, μ̄, χ̄) holds.

2. The truth value of Unif(λ, S, μ̄, χ̄) is invariant under nonstationary changes to χ̄.

3. The truth value of Unif(λ, S, μ̄, χ̄) is invariant under nonstationary changes to S.

4. If Unif(λ, S, μ̄, χ̄) holds, it holds for any smaller (≤) χ-sequence and any larger (≥) μ-sequence.

5. Unif(λ, S, μ̄, χ̄) implies that D_{λ}(χ̄) has at most |D_{λ}(μ̄)| many equivalence classes under equivalence modulo the nonstationary ideal restricted to S.

6. Unif(λ, S, μ̄, χ̄) follows from the statement that for some β < λ, D_{λ}(χ̄) has at most |D_{β}(μ̄)| many equivalence classes under the nonstationary ideal restricted to S.

The first major result of the appendix is the following (Shelah’s Lemma 1.5), where for a sequence μ̄ = ⟨μ̄(i) : i < λ⟩ and an α < λ we let μ̄[α] be the sequence of length λ whose ith element is μ̄(α + i).

Theorem 0.7. Given λ, S, μ̄ and λ̄ as above, and letting μ_{0} = |D(μ̄)| and μ_{1} = min_{α<λ} |D(μ̄[α])|, Unif(λ, S, μ̄, χ̄) and Unif(λ, S, μ_{0}, μ_{1}, χ̄) are equivalent.

Note that μ_{0} ≥ μ_{1} in the statement of Theorem 0.7. The first lemma (Fact 1.5A) towards the proof of Theorem 0.7 involves a change of μ-sequence.

Lemma 0.8. Suppose that μ̄ and ν̄ are sequences of nonzero cardinals of length λ, and suppose that there is a continuous, injective, order-preserving partial map g: D(μ̄) → D(ν̄) such that for every η ∈ D_{λ}(μ̄) the set of i < λ with η↾i ∈ Dom(g) is club. Then Unif(λ, S, μ̄, χ̄) implies Unif(λ, S, ν̄, χ̄).

Proof of Lemma 0.8. First note that we may assume that g is length-preserving as well. Given F witnessing Unif(λ, S, μ̄, χ̄), define F^{∗}(η) to be F (g^{−1}(η)) for all η in the range of g, and let F^{∗} be defined in any manner on other η’s. Then F^{∗} witnesses Unif(λ, S, ν̄, χ̄).

Proof of the forward direction of Theorem 0.7. Let α < λ be such that μ_{1} = |D(μ̄[α])|. Then μ̄(i) ≤ μ_{1} for all i ∈ [α, λ). Let μ′_{0} = |D_{α}(μ̄)|, and let ⟨ν_{ξ} : ξ < μ′_{0}⟩ enumerate D_{α}(μ̄). Then μ′_{0} ≤ μ_{0}. Then there is a partial function g: D(μ̄) → D(μ_{0}, μ_{1}) as in the statement of Lemma 0.8 with domain the set of η ∈ D(μ̄) of length at least α, defined by g(ν_{ξ}^{⌢}η) = ⟨ξ⟩^{⌢}η.

For the reverse direction of Theorem 0.7, Shelah proves the following lemmas (the second and third of which make up Claim 1.6), which show that it suffices to suppose that ⟨μ̄(i) : 1 ≤ i < λ⟩ is nondecreasing.

Lemma 0.9. If there exists j < λ such that μ̄(i) = 1 for all i ∈ [j, λ), then μ_{0} = |D_{j}(μ̄)|, μ_{1} = 1, and the conclusion of Theorem 0.7 holds.

Proof. By Lemma 0.6, parts (5) and (6).

Lemma 0.10. If ⟨α_{i} : i < λ⟩ is an increasing, continuous sequence of elements of λ with α_{0} = 0, and for each i < λ, ν_{i} = |Π_{α_{i}≤j<α_{i+1}} μ̄(j)|, then Unif(λ, S, μ̄, χ̄) and Unif(λ, S, ν̄, χ̄) are equivalent.

Proof. Translate between F ’s using the natural bijection between D(μ̄) and D(ν̄).

Lemma 0.11. There exists ⟨α_{i} : i < λ⟩, an increasing, continuous sequence of elements of λ with α_{0} = 0, such that, letting ν_{i} = |Π_{α_{i}≤j<α_{i+1}} μ̄(j)| for each i < λ, ⟨ν_{i} : 1 ≤ i < λ⟩ is nondecreasing.

Proof. Let κ^{∗} be the least cardinal κ such that {i < λ | μ̄(i) ≥ κ} is bounded below λ. Let α^{∗} < λ be such that μ̄(i) < κ^{∗} for all i ∈ [α^{∗}, λ). Let α_{1} = α^{∗}. There are three cases, depending on whether κ^{∗} is a successor cardinal, has cofinality λ, or cofinality less than λ.

Proof of the reverse direction of Theorem 0.7. By Lemmas 0.9, 0.10 and 0.11, we may assume that μ̄(i) ≤ μ̄(j) whenever 1 ≤ i ≤ j < λ. By Lemma 0.8, we need only find a partial injective, continuous, order-preserving embedding g from D(μ_{0}, μ_{1}) to D(μ̄) with the property that every η ∈ D_{λ}(μ_{0}, μ_{1}) has club many initial segments in the domain of g. Let α^{∗} ∈ [3, λ) be such that for all β ∈ [α^{∗}, λ), |D(μ̄[β])| = μ_{1}. The desired function g can be defined in a straightforward recursive manner once one sees that D(μ̄) contains an antichain of size μ_{0} (consisting of elements of length at least α^{∗}, and that there is an antichain of size μ_{1} extending each element of D(μ̄) of length at least α^{∗}. This construction is relatively straightforward, alternating codes and 0’s until the coding is done, then punctuating with two 1’s. In the first case, one starts with ν(0) and fills until α^{∗} with all 0’s. Since μ̄ is increasing, one can always save values for later. The second case is simpler.

The following lemma (Shelah’s Claim 1.7 (2)) says that fewer than λ many witnesses to Unif(λ, S, …) can be glued together to make a witness to Unif(λ, S, …) on the corresponding product. Shelah’s Claim 1.7 (1) is a special case of Claim 1.7 (2).

Lemma 0.12. Suppose that κ < λ, and Unif(λ, S, μ̄_{ξ}, χ̄_{ξ}) holds for all ξ < κ. Define μ̄(i) and χ̄(i) for i < λ by μ̄(i) = |Π_{ξ<κ}μ̄_{ξ}(i)| and χ̄(i) = |Π_{ξ<κ}χ̄_{ξ}(i)|. Then Unif(λ, S, μ̄, χ̄) holds.

Proof. Fix witnesses F_{ξ} (ξ < κ) to Unif(λ, S, μ̄_{ξ}, χ̄_{ξ}), and fix bijections G^{i}: μ̄(i) → Π_{ξ<κ}μ̄_{ξ}(i) and H^{i} : Π_{ξ<κ}μ̄_{ξ}(i) → χ̄(i). For each i < λ, ξ < κ and α < μ̄(i), G_{ξ}^{i}(α) be the ξth coordinate of G^{i}(α), and let H_{ξ}^{i}(α) be the ξth coordinate of H^{i}(α). Given η ∈ D_{δ}(μ̄) for some δ < λ, let

F (η) = H^{δ}(⟨F_{ξ}(⟨G_{ξ}^{i}(η(i)) : i < δ⟩) : ξ < κ⟩).

Now given h ∈ D_{λ}(χ̄), let h_{ξ} (ξ < κ) be such that H^{δ}(⟨h_{ξ}(δ) : ξ < κ⟩) = h(δ) for all δ < λ. Fix η_{ξ} (ξ < κ) and clubs C_{ξ} (ξ < κ) such that for each ξ < κ and each δ ∈ C_{ξ} ∩ S, F_{ξ}(η_{ξ}↾δ) = h_{ξ}(δ). Let η ∈ D(μ̄) be such that G^{i}(η(i)) = ⟨η_{ξ}(i) : ξ < κ⟩ for all i < λ. Then for every δ ∈ S ∩ ⋂_{ξ<κ} C_{ξ},

F (η↾δ) = H^{δ}(⟨F_{ξ}(⟨G_{ξ}^{i}(η(i)) : i < δ⟩) : ξ < κ⟩) =
H^{δ}(⟨F_{ξ}(η_{ξ}↾δ) : ξ < κ⟩) = H^{δ}(⟨h_{ξ}(δ) : ξ < κ⟩) = h(δ).

The following is Claim 1.7 (3).

Lemma 0.13. Suppose that μ̄ is a nondecreasing sequence of infinite cardinals, Unif(λ, S, μ̄, χ̄) holds, and ρ̄(i) ≤ χ̄(i)^{|i|} for all i < λ. Then Unif(λ, S, μ̄, ρ̄) holds.

Proof. It suffices to suppose that ρ̄(i) = ξ̄(i)^{|i|} for all i < λ, by Lemma 0.6 (4). Let F witness Unif(λ, S, μ̄, χ̄). For each i < λ, fix a bijection H^{i}: Π_{j<i}ξ̄(i) → ρ̄(i). Fix increasing functions k_{ζ}: λ → λ (ζ < λ) with disjoint ranges. For any η ∈ D(μ̄) ∪ D_{λ}(μ̄) and any ζ < λ, let η[ζ] be the sequence of values that η takes on the range of k_{ζ}.

Let C^{∗} be the set of δ < λ such that the range of each k_{ζ} (ζ < δ) has ordertype δ below δ. Then C^{∗} is a club. Given δ ∈ C^{∗}, let E(δ) be the set of η ∈ D_{δ}(μ̄) such that for each ζ < δ, η[ζ] ∈ D_{δ}(μ̄). For each η ∈ E(δ), let

F^{∗}(η) = H^{δ}(⟨F (η[ζ]) : ζ < δ⟩).

Then given h ∈ D_{λ}(ρ̄), for each ζ < λ, let h_{ζ} be the function with domain (ζ, λ) such that for each δ < λ, H^{δ}(⟨h_{ζ}(δ) : ζ < δ⟩) = h(δ). Then for each ζ < λ there exist an η_{ζ} ∈ D_{λ}(μ̄ and a club C_{ζ} such that for all δ ∈ C_{ζ}, F (η_{ζ}↾δ) = h_{ζ}(δ). Let η ∈ D_{λ}(μ̄) be such that the sequence of values η gives on the range of each k_{ζ} is equal to η_{ζ}. Then if δ is in C^{∗} and in the diagonal intersection of the C_{ζ}’s, then

F^{∗}(η↾δ) = H^{δ}(⟨F ((η↾δ)[ζ]) : ζ < δ⟩) =
H^{δ}(⟨F (η_{ζ}↾δ) : ζ < δ⟩) = H^{δ}(⟨h_{ζ}(δ) : ζ < δ⟩) = h(δ).

In his Conclusion 1.8, Shelah mentions that if Unif(λ, μ(0), 2, χ) holds, and 1 < κ < λ and μ(0)^{κ} = μ(0), then by Lemma 0.12 we have Unif(λ, μ(0), 2^{κ}, χ^{κ}), which by our Lemma 0.7 applied twice is equivalent to Unif(λ, μ(0), 2, χ^{κ}), since the corresponding μ_{0} and μ_{1} are the same in each case.

In his Lemma 1.9, (1), Shelah mentions that Id-Unif(λ, μ̄, χ̄) is either all of P(λ) or an ideal. To see this, given F_{0} and F_{1} which work for S_{0} and S_{1}, use F_{0} on sequences whose lengths are in S_{0}, and F_{1} for sequences whose lengths are in S_{1}.

The first half of Shelah’s Remark 1.9A notes that Id-Unif(λ, μ̄, χ̄) is equivalent to Id-Unif(λ, μ_{0}, μ_{1}, χ̄), by our Lemma 0.7. The second half of the remark uses later material, so we will save it for later (after Lemmas 0.19-0.23).

The second half of Shelah’s Lemma 1.9 is the following.

Lemma 0.14. If μ̄ is nondecreasing then Id-Unif(λ, μ̄, χ̄) is closed under diagonal unions.

Proof. Let S_{i} (i ∈ λ) be elements of Id-Unif(λ, μ̄, χ̄) as witnessed by functions F_{i} (i < λ). Let S be the diagonal union of ⟨S_{i} : i < λ⟩, and let f: S → λ be a regressive function such that δ ∈ S_{f(δ)} for each δ ∈ S. Let ⟨k_{ζ} : ζ < λ⟩ and C^{∗} be as in the proof of Lemma 0.13. For each δ ∈ C^{∗} and η ∈ D_{δ}(μ̄), let F (η) = F_{f(δ)}(η[f(δ)]), where η[ζ] is defined as in Lemma 0.13. Now given h ∈ D(χ̄), for each ζ < λ, let η_{ζ} ∈ D_{λ}(μ̄) and club C_{ζ} ⊆ λ be such that F_{ζ}(η_{ζ}↾δ) = h(δ) for all δ ∈ C_{ζ} ∩ S_{ζ}. Let η ∈ D_{λ}(μ̄) be such that η[ζ] = η_{ζ} for all ζ < λ. Then for all δ ∈ S ∩ C^{∗} ∩ △_{ζ<λ}C_{ζ},

F (η↾δ) = F_{f(δ)}((η↾δ)[f(δ)]) =
F_{f(δ)}(η_{f(δ)}↾δ) = h(δ).

Now we are finally up to Shelah’s Theorem 1.10.

Theorem 0.15. If λ is regular, 2^{<λ} < 2^{λ} and there is no collection of sets S_{i} ∈ [μ]^{λ} (i < 2^{λ}) with pairwise finite intersection, then Unif(λ, μ, 2^{<λ}, 2^{<λ}) fails.

Shelah notes that the existence of such a collection of S_{i}’s would imply that μ^{ℵ_{0}} ≥ 2^{λ}, by picking a countable subset of each S_{i}. He notes that the nonexistence of such a collection implies that μ < 2^{λ}. He also notes that his Theorem 1.10 has the following corollary.

Corollary 0.16. If for some θ < λ, 2^{θ} = 2^{<λ} < 2^{λ}, then Unif(λ, 2^{θ}, 2) fails.

Proof of Corollary 0.16. By Theorem 0.15 we get that Unif(λ, 2^{θ}, 2^{θ}, 2^{θ}) fails. This is equal to Unif(λ, 2^{θ}, 2^{θ}, 2) by Lemmas 0.6 (2) and 0.12.

Proof of Theorem 0.15. Let F witness Unif(λ, μ, 2^{<λ}, 2^{<λ}). Let Mod be the set of sequences

⟨α, C_{0}, g_{0}, C_{1}, g_{1}, … C_{β}, g_{β}, …⟩_{β<β(0)}

where α and β(0) are elements of λ, each g_{β} is a function from α \ {0} to ^{<λ}2, and each C_{β} is a closed subset of α. Then Mod has cardinality 2^{<λ}, so we can fix a bijection H from Mod to ^{<λ}2. For each function f ∈ ^{λ}2 and each β < λ, define h_{f,β}: λ → ^{<λ}2, g_{f,β} ∈ D_{λ}(μ, 2^{<λ}) and club C_{f,β} ⊆ λ as follows.

• h_{f,0} = g_{f,0} = f and C_{f,0} = λ \ {0};

• for γ > 0:

– h_{f,γ}(i) =

H(⟨α, C_{f,0} ∩ α, g_{f,0}↾(α \ {0}, …, C_{f,β} ∩ α, g_{f,β}↾(α \ {0}), …⟩_{β<γ},

where α = α(i, f, γ) = min ⋂_{β<γ} C_{f,β} \ (i + 1);

– g_{f,γ} is such that some club C is subset of

{δ < λ | F (g_{f,γ}↾δ) = h_{f,γ}(δ)},

and C_{f,γ} is the set of limit points of C ∩ ⋂_{β<γ} C_{f,β}.

The key claim is the following: if f_{1} and f_{2} are distinct functions from λ to 2, and f_{1}(0) = f_{2}(0), then the set of γ < λ such that

g_{f_{1},γ}↾ min C_{f_{1},γ} = g_{f_{2},γ}↾ min C_{f_{2},γ}

is finite.

Given the claim, the remainder of the proof is as follows. For each f: λ → 2, let A_{f} be the set of sequences ⟨γ, g_{f,γ}(0), g_{f,γ}↾(δ \ {0}), f(0)⟩, where γ < λ and δ = min C_{f,γ}. For distinct f’s, these sets have finite intersection, so there are 2^{λ} such A_{f}’s. Each A_{f} is a subset of λ × μ ×^{<λ}(^{<λ}2) × 2 of cardinality λ. If μ ≥ 2^{<λ} then we have a contradiction. Otherwise, 2^{λ} is less than or equal to (2^{<λ})^{ℵ_{0}} (since the A_{f}’s have distinct countable subsets), which is equal to 2^{<λ}, giving another contradiction. This finishes the proof of the theorem, assuming the claim.

Finally, we prove the claim. To do this, suppose that f_{1} and f_{2} are functions from λ to 2 such that f_{1}(0) = f_{2}(0), and let ⟨j_{n} : n < ω⟩ be an increasing sequence of members of λ, with supremum j, such that

g_{f_{1},j_{n}}↾ min C_{f_{1},j_{n}} = g_{f_{2},j_{n}}↾ min C_{f_{2},j_{n}}

for all n < ω. For each ℓ ∈ {1, 2}, let C^{ℓ} = ⋂_{n<ω} C_{f_{ℓ},j_{n}} and let ⟨γ_{i}^{ℓ} : i < λ⟩ be an increasing enumeration of C^{ℓ}. For notational convenience, let γ_{λ}^{ℓ} = λ for ℓ ∈ {1, 2}.

We prove by induction on i ≤ λ that a) γ_{i}^{1} = γ_{i}^{2};

b) for all ζ < j,

g_{f_{1},ζ}↾(γ_{i}^{1} \ {0}) = g_{f_{2},ζ}↾(γ_{i}^{2} \ {0})

and

C_{f_{1},ζ} ∩ γ_{i}^{1} = C_{f_{2},ζ} ∩ γ_{i}^{2}.

Letting i = λ and ζ = 0, this shows that f_{1} = f_{2}, since g_{f,0} = f for all f ∈ ^{λ}2.

The induction has three cases: i = 0, i limit and i successor. For each n ∈ ω we let δ_{n} = min Cf_{1}, j_{n} = min C_{f_{2},j_{n}}.

Case i = 0. For each ℓ ∈ {1, 2}, ⟨C_{f_{ℓ},j_{n}} : n < ω⟩ is a ⊆-decreasing sequence of clubs, so γ_{0}^{ℓ} = sup{δ_{n} : n < ω}. This shows (a). To see (b), fix ζ < j and n ∈ ω such that ζ < ȷ_{n} and if

g_{f_{1},ζ}↾(γ_{0}^{1} \ {0}) ≠ g_{f_{2},ζ}↾(γ_{0}^{2} \ {0})

then

g_{f_{1},ζ}↾(δ_{n} \ {0}) ≠ g_{f_{2},ζ}↾(δ_{n} \ {0})

and if

C_{f_{1},ζ} ∩ γ_{0}^{1} ≠ C_{f_{2},ζ} ∩ γ_{0}^{2}

then

C_{f_{1},ζ} ∩ δ_{n} ≠ C_{f_{2},ζ} ∩ δ_{n}.

We have assumed that g_{f_{1},j_{n}}↾δ_{n} = g_{f_{2},j_{n}}↾δ_{n}. Also, for each ℓ ∈ {1, 2}, δ_{n} ∈ C_{f_{ℓ},j_{n}}, so

F (g_{f_{ℓ},j_{n}}↾δ_{n}) = h_{f_{ℓ},j_{n}}(δ_{n}) = H(⟨α, …, C_{f_{ℓ},β} ∩ α, g_{f_{ℓ},β}↾(α \ {0}), …⟩_{β<j_{n}}),

where

α = α(δ_{n}, f_{ℓ}, j_{n}) = min[ ⋂_{β<j_{n}} C_{f_{ℓ},β} \ (δ_{n} + 1)].

As H is injective, it follows that

g_{f_{1},ζ}↾(δ_{n} \ {0}) = g_{f_{2},ζ}↾(δ_{n} \ {0})

and

C_{f_{1},ζ} ∩ δ_{n} = C_{f_{2},ζ} ∩ δ_{n},

which means that

g_{f_{1},ζ}↾(γ_{0}^{1} \ {0}) = g_{f_{2},ζ}↾(γ_{0}^{2} \ {0})

and

C_{f_{1},ζ} ∩ γ_{0}^{1} = C_{f_{2},ζ} ∩ γ_{0}^{2}.

This concludes the proof of the case i = 0.

The case where i is a limit ordinal is immediate.

Finally, suppose that (a) and (b) hold for i < λ, and let us see that they hold for i + 1. We have that for all n ∈ ω, g_{f_{1},j_{n}}↾γ_{0}^{1} = g_{f_{2},j_{n}}↾γ_{0}^{2}. Note that γ_{0}^{1} and γ_{0}^{2} are both nonzero, as 0 ∉ C_{f,0} for all functions f ∈ ^{λ}2. By the induction hypothesis,

g_{f_{1},j_{n}}↾(γ_{i}^{1} \ {0}) = g_{f_{2},j_{n}}↾(γ_{i}^{2} \ {0}).

Putting these two facts together, we have that

g_{f_{1},j_{n}}↾γ_{i}^{1} = g_{f_{2},j_{n}}↾γ_{i}^{2}.

For ℓ ∈ {1, 2} we have γ_{i}^{ℓ} ∈ ⋂_{n∈ω} C_{f_{ℓ},j_{n}}, so for all n ∈ ω, F (g_{f_{ℓ},j_{n}}↾γ_{i}^{ℓ}) = h_{f_{ℓ},j_{n}}(γ_{i}^{ℓ}), which is equal to

H(⟨α_{n}^{ℓ}, …, C_{f_{ℓ},β} ∩ α_{n}^{ℓ}, g_{f_{ℓ},β}↾(α_{n}^{ℓ} \ {0}), …⟩_{β<j_{n}}),

where α_{n}^{ℓ} = α(γ_{i}^{ℓ}, f_{ℓ}, j_{n}) = min[⋂_{β<j_{n}} C_{f_{ℓ},β} \ (γ_{i}^{ℓ} + 1)]. As H is injective, it follows that α_{n}^{1} = α_{n}^{2} for each n ∈ ω. Since for each f ∈ ^{λ}2 and each β < β′ < λ, C_{f,β′} is contained in the limits points of C_{f,β}, it follows that ⟨α_{n}^{ℓ} : n < ω⟩ is increasing for each ℓ ∈ {1, 2}. It follows furthermore (for the same reason) that for each ℓ ∈ {1, 2},

⋃_{n∈ω} α_{n}^{ℓ} = min[ ⋂_{β<j_{n}} C_{f_{ℓ},β} \ (γ_{i}^{ℓ} + 1)] = γ_{γ+1}^{ℓ},

so γ_{i+1}^{1} = γ_{i+1}^{2} (which is part (a)). Clause (b) follows from the fact that every ζ < j is less than some j_{n}, and the fact that γ_{i+1}^{1} = γ_{i+1}^{2} is the supremum of the set of ordinals α_{n}^{1} = α_{n}^{2} for n ∈ ω.

0.17 Definition. Given a set X and a cardinal λ, a collection F ⊆ [X]^{λ}, is called a (X, λ)-cover if every subset of X of cardinality λ has a superset in F. We let cov(X, λ) denote the least cardinality of a (X, λ)-cover consisting of sets of cardinality λ.

The following facts are immediate, for sets X and Y and cardinals λ and μ.

1. X ⊆ Y ⇒ cov(X, λ) ≤ cov(Y, λ).

2. |X| ≤ |Y | ⇒ cov(X, λ) ≤ cov(Y, λ).

3. |X| = |Y | ⇒ cov(X, λ) = cov(Y, λ).

4. λ < μ ⇒ cov(μ, λ) ≥ μ.

5. cov(λ, λ) = 1.

Lemma 0.18. Suppose that λ and μ are cardinals, and α is an ordinal.

1. if λ ≤ μ, then cov(μ^{+}, λ) =cov(μ, λ) + μ^{+}.

2. if μ is a limit ordinal, λ < μ and ⟨μ_{i} : i < cf(μ)⟩ is an increasing sequence with limit μ and μ_{0} > λ, then cov(μ, λ) ≤ Π_{i<cf(μ)}cov(μ_{i}, λ).

3. cov(λ^{+α}, λ) ≤ (λ^{+α})^{|α|}.

Proof. For the first conclusion, ≥ follows from facts (1) and (4) listed above. The other direction follows from the fact that each subset of μ^{+} of cardinality λ is bounded. For the second conclusion, take a product of the covers for each μ_{i}. For the third conclusion, argue by induction on α. The case α = 0 is fact (5) above. To go from β to β + 1, note that cov(λ^{β+1}, λ) =cov(λ^{β}, λ) + λ^{β+1} (by the first conclusion of this lemma) which is less than or equal to (λ^{β})^{|β|} + λ^{β+1} (by the induction hypothesis) which is less than or equal to (λ^{β+1})^{β+1}. If α is a limit of the sequence ⟨α_{i} : i < cf(α)⟩, then cov(λ^{+α}, λ) ≤ Π_{i<cf(α)}cov(λ^{+α_{i}}, λ) (be the second conclusion of this lemma) which is less than or equal to ((λ^{+α})^{|α|})^{cf(α)} = (λ^{+α})^{|α|}.

The next five lemmas are the first five parts of Shelah’s Lemma 1.14, revised to accommodate the sixth part. In all cases, I believe I have used a hypothesis no stronger than the one used by Shelah.

Lemma 0.19. Suppose that λ and μ are cardinals, with λ regular and uncountable, and let χ̄ = ⟨χ_{i} : i < λ⟩ be a sequence of nonzero cardinals. Let S be a subset of λ, and suppose that Unif(λ, S, μ, χ̄) holds. Then Unif(λ, S, cov(μ, λ), λ, χ̄) holds.

Proof. Let F witness Unif(λ, S, μ, χ̄), let ⟨A_{i} : i < cov(μ, λ)⟩ be a (μ, λ)-cover, and enumerate each A_{i} by ⟨α_{i,j} : j < λ⟩. For each η ∈ D(cov(μ, λ), λ), let

F^{∗}(η) = F (⟨α_{η(0),η(1+i)} : i < length(η)⟩).

Then given h ∈ D_{λ}(χ̄), let η ∈ D_{λ}(μ) be such that {i ∈ S : F (η↾i) = h(i)} contains a club relative to S. Let η^{∗} ∈ D_{λ}(cov(μ, λ), λ) be such that range(η) ⊆ A_{η^{∗}(0)} and such that η^{∗}(1+i) = η(i). Then F^{∗}(η^{∗}↾i) = F (η↾i) for all i ≥ ω.

Lemma 0.20. Suppose that λ, χ, μ_{0} and μ_{1} are cardinals, with λ regular and uncountable, χ ≥ λ and cov(χ, λ) ≤ μ_{0}. Let S be a subset of λ. Then Unif(λ, S, μ_{0}, μ_{1}, χ) holds if and only if Unif(λ, S, μ_{0}, μ_{1}, λ) holds.

Proof. The forward direction follows from Lemma 0.6 (4). For the reverse direction, let F witness Unif(λ, S, μ_{0}, μ_{1}, λ). Let e_{0} and e_{1} be functions from μ_{0} to μ_{0} such that α ↦ (e_{0}(α), e_{1}(α)) is a bijection from μ_{0} to μ_{0} × μ_{0}. Let ⟨A_{i} : i < μ_{0}⟩ be a (χ, λ)-cover, and for each i < μ_{0} let q_{i}: λ → A_{i} be a bijection. For each η ∈ D(μ_{0}, μ_{1}), let k(η) be the sequence produced by replacing the first member of η with e_{1}(η(0)), and let F^{∗}(η) = q_{e_{0}(η(0))}(F (k(η))). Then given h ∈ D(χ), let i < λ be such that the range of h is contained in A_{i}. Then there is an η ∈ D_{λ}(μ_{0}, μ_{1}) such that F (η↾i) = q_{i}^{−1}(h(i)) for a club of i relative to S. Let η^{∗} ∈ D_{λ}(μ_{0}, μ_{1}) be such that e_{0}(η^{∗}(0)) = i, e_{1}(η^{∗}(0)) = η(0) and η^{∗}(j) = η(j) for all nonzero j in the domain of η. Then for all j < λ, F^{∗}(η^{∗}↾j) = q_{i}(F (k(η^{∗}↾j))), which is equal to h(j) for a club of j relative to S.

Lemma 0.21. Suppose that λ, χ, μ_{0} and μ_{1} are cardinals, with λ regular and uncountable, χ ≥ λ and cov(χ, λ) ≤ μ_{0}. Suppose that λ is not a strong limit cardinal, and that χ ≥ 2. Let S be a subset of λ. Then Unif(λ, S, μ_{0}, μ_{1}, χ) holds if and only if Unif(λ, S, μ_{0}, μ_{1}, 2) holds.

Proof. The forward direction follows from Lemma 0.6 (4). For the reverse direction, it suffices to consider the case χ = λ, by Lemma 0.20. Since λ is not a strong limit, this case follows from Lemma 0.13 and parts (2) and (4) of Lemma 0.6.

Lemma 0.22. Suppose that λ, μ_{0} and μ_{1} are cardinals, with λ regular and uncountable, and λ ≤ μ_{1}. Let χ̄ = ⟨χ_{i} : i < λ⟩ be a sequence of nonzero cardinals. Let S be a subset of λ, and suppose that Unif(λ, S, μ_{0}, μ_{1}, χ̄) holds. Then Unif(λ, S, μ_{0} + cov(μ_{1}, λ), λ, χ̄) holds.

Proof. If μ_{0} ≥ cov(μ_{1}, λ) then this follows from Lemma 0.6 (4). Supposing otherwse, μ_{0} + cov(μ_{1}, λ) = μ_{0} · cov(μ_{1}, λ) = cov(μ_{1}, λ), so we may fix functions e_{0} : cov(μ_{1}, λ) → μ_{0} and e_{1} : cov(μ_{1}, λ) → cov(μ_{1}, λ) such that α ↦ (e_{0}(α), e_{1}(α)) is a bijection from cov(μ_{1}, λ) to μ_{1} × cov(μ_{1}, λ). Let F witness Unif(λ, S, μ_{0}, μ_{1}, χ̄), let ⟨A_{i} : i < cov(μ_{1}, λ)⟩ be a (μ_{1}, λ)-cover, and enumerate each A_{i} by ⟨α_{i,j} : j < λ⟩. For each η ∈ D(cov(μ, λ), λ), k(η) be the sequence obtained by replacing η(0) with e_{0}(η(0)), and η(i) for each nonzero i with α_{e_{1}(η(0)),η(i)}, and let

F^{∗}(η) = F (k(η)).

Then given h ∈ D_{λ}(χ̄), let η ∈ D_{λ}(μ_{0}, μ_{1}) be such that {i ∈ S : F (η↾i) = h(i)} contains a club relative to S.

Let i < λ be such that the range of η is contained in A_{i}. Let η^{∗} ∈ D_{λ}(μ_{0} + cov(μ_{1}, λ), μ_{1}) be such that e_{1}(η^{∗}(0)) = i, e_{0}(η^{∗}(0)) = η(0) and α_{i,η^{∗}(j)} = η(j) for all nonzero j in the domain of η. Then for all j < λ, F^{∗}(η^{∗}↾j) = F (k(η^{∗}↾j)), which is equal to h(j) for a club of j relative to S.

Lemma 0.23. Suppose that λ, μ_{0} and μ_{1} are cardinals, with λ regular, uncountable and not a strong limit. Let χ̄ = ⟨χ_{i} : i < λ⟩ be a sequence of nonzero cardinals. Suppose that μ_{0} ≥ cov(μ_{1}, λ) and λ < μ_{1}. Let S be a subset of λ, and suppose that Unif(λ, S, μ_{0}, μ_{1}, χ̄) holds. Then Unif(λ, S, μ_{0}, 2, χ̄) holds.

Proof. The reverse direction follows from Lemma 0.6 (4). For the forward direction, first note that Lemma 0.22 gives us Unif(λ, μ_{0}, λ, χ̄) from Unif(λ, μ_{0}, μ_{1}, χ̄). Then Unif(λ, μ_{0}, 2, χ̄) follows from Lemma 0.7, since the two “μ_{1}”’s are the same in this case (in checking this fact, it may help to break into cases, depending on whether λ is a strong limit or not).

Let us return now to the second part of Shelah’s Remark 1.9A. We want to see that Unif(λ, S, μ_{0}, μ_{1}, χ̄) ⇔ Unif(λ, S, μ_{0}, μ_{0}, χ̄), when μ_{0} and μ_{1} are as in Lemma 0.7 and cov(μ_{0}, λ) = μ_{0}. Since μ_{0} ≥ μ_{1}, the forward direction follows from Lemma 0.6 (4). For the other direction, Unif(λ, S, μ_{0}, μ_{0}, χ̄) implies Unif(λ, μ_{0}, 2, χ̄) by Lemma 0.23, and Unif(λ, μ_{0}, 2, χ̄) implies Unif(λ, μ_{0}, μ_{1}, χ̄) by Lemma 0.6 (4).

The following is Shelah’s Conclusion 1.15.

Conclusion 0.24. Suppose that μ is a cardinal less than ℵ_{ω_{1}} and that μ^{ℵ_{0}} < 2^{ℵ_{1}}. Then Unif(ω_{1}, μ, μ, 2) fails.

Proof. Suppose that Unif(ω_{1}, μ, μ, 2) holds. Then by Lemmas 0.13 and 0.6(2), Unif(ω_{1}, μ, μ, 2^{ℵ_{0}}) holds. Then by Lemma 0.19, Unif(ω_{1}, cov(μ, ℵ_{1}), ℵ_{1}, 2^{ℵ_{0}}) holds. Let α < ω_{1} be such that μ = ω_{α}. Then by Lemma 0.18 (3),

cov(μ, ω_{1}) ≤ ℵ_{α}^{|α|} ≤ μ^{ℵ_{0}} < 2^{ℵ_{1}},

so we can apply Theorem 0.15 with “μ” as cov(μ, ω_{1}), getting the failure of Unif(ω_{1}, cov(μ, ω_{1}), 2^{ℵ_{0}}, 2^{ℵ_{0}}) (note that 2^{<ℵ_{1}} = 2^{ℵ_{0}}) and thereby a contradiction.

Theorem 0.26 below is Shelah’s Theorem 1.16, with the hypothesis μ > λ dropped. As we see in the second line of the proof, this hypothesis follows from the other assumptions of the theorem. With this hypothesis removed, Theorem 0.26 is a strengthening our Theorem 0.15 The proof is a modification of the proof of Theorem 0.15.

0.25 Definition. Let (∗)_{2^{λ},μ,λ^{+}} denote the statement that there exists a family {S_{i} :< 2^{λ}} of elements of [μ]^{λ^{+}} such that |S_{i} ∩S_{j}| < ℵ_{0} for all distinct i, j < 2^{λ}.

Theorem 0.26. Suppose that μ and λ are cardinals, with λ a regular uncountable cardinal. Suppose that 2^{<λ} < 2^{λ} and that Unif(λ, μ, 2^{<λ}, 2^{<λ}) holds. Then (∗)_{2^{λ},μ,λ^{+}} holds.

Proof. As before, the conclusion is immediate if μ ≥ 2^{λ}, so assume otherwise. Similarly, by Theorem 0.15, we have that 2^{λ} ≤ μ^{ℵ_{0}}. Since (2^{<λ})^{ℵ_{0}} = 2^{<λ} < 2^{λ}, it follows that μ > 2^{<λ}.^{1}

For each α < λ^{+}, let ⟨B_{i}^{α} : i < λ⟩ be a continuous, ⊆-increasing sequence of sets of cardinality less than λ, with union α.

Let Mod be the set of sequences

⟨α, …, … C_{β}, g_{β}, …⟩_{β∈B_{i}^{α}}

for some i < λ, where each g_{β} is a function from B_{i}^{α} \ {0} to ^{<λ}2, and each C_{β} is a closed subset of i. Note that one can recover the corresponding α and i from any given member of Mod. Furthermore, Mod has cardinality 2^{<λ}, so we can fix a bijection H from Mod to ^{<λ}2.

Let F witness Unif(λ, μ, 2^{<λ}, 2^{<λ}). For each function f ∈ ^{λ}2 we can define by recursion on β < λ^{+} functions h_{f,β}: λ → ^{<λ}2, g_{f,β} ∈ D_{λ}(μ, 2^{<λ}) and club sets C_{f,β} ⊆ λ as follows.

1The proof contains the following observation, which is interesting but doesn’t appear to be necessary: By Hausdorff’s formula, if κ is a cardinal and n is an integer, then (κ^{+n})^{ℵ_{0}} = κ^{+n} + κ^{ℵ_{0}}. If μ = (2^{<λ})^{n} for some integer n, then,

μ^{ℵ_{0}} = μ + (2^{<λ})^{ℵ_{0}} = μ + 2^{<λ} = μ < 2^{λ} ≤ μ^{ℵ_{0}}, giving a contradiction. So μ ≥ (2^{<λ})^{+ω}.

• h_{f,0} = g_{f,0} = f and C_{f,0} = λ \ {0};

• for λ > 0:

– h_{f,γ}(i) = H(⟨α, …, C_{f,β} ∩ α, g_{f,β}↾(α \ {0}), …⟩_{β∈B_{i}^{γ}}, where α = α(i, f, γ) = min ⋂_{β∈B_{i}^{γ}} C_{f,β} \ (i + 1);

– g_{f,γ} is such that some club C is subset of

{δ < λ | F (g_{f,γ}↾δ) = h_{f,γ}(δ)},

and C_{f,γ} is the set δ which are of limits points of C and of every C_{f,β} with β ∈ B_{δ}^{γ}.

The key claim is the following: if f_{0} ∈ ^{λ}2 and ⟨j_{n} : n < ω⟩ is an increasing sequence of elements of λ, then

|{f ∈ ^{λ}2 : ∀n ∈ ω g_{f,j_{n}}(0) = g_{f_{0},j_{n}}(0)}| ≤ 2^{<λ}.

Given the claim, the remainder of the proof is as follows. For each f_{0} : λ → 2, let

Y′_{f_{0}} = {f ∈ ^{λ}2 : |{j < λ^{+} : g_{f,j}(0) = g_{f_{0},j}(0)}| ≥ ℵ_{0}}.

The set of increasing ω-sequences from λ^{+} has cardinality (λ^{+})^{ℵ_{0}}, which (since λ is regular and uncountable) is at most 2^{<λ} + λ^{+}, which (since λ < μ < 2^{λ}) is less than 2^{λ}. Furthermore, for each f_{0} ∈ ^{λ}2, |Y′_{f_{0}}| has cardinality at most (λ^{+})^{ℵ_{0}} × 2^{<λ}, which is also less than 2^{λ}.^{2} The set of pairs (f_{0}, f_{1}) ∈ ^{λ}2 such that f_{0} ∈ Y′_{f_{1}} forms an equivalence relation on^{λ}2. Let F^{∗} be a subset of^{λ}2 which intersects each equivalence class in exactly one point. Since each equivalence class has cardinality less than 2^{λ}, |F^{∗}| = 2^{λ}. Then

{{(g_{f,j}(0), j) : j < λ^{+}} : f ∈ F^{∗}}

is a family of 2^{λ} many subsets of μ×λ^{+} of cardinality λ^{+}, having pairwise finite intersection. This finishes the proof of the theorem, assuming the claim.

Finally, we prove the claim. To do this, suppose towards a contradiction that f_{0} ∈ ^{λ}2 and ⟨j_{n} : n ∈ ω⟩ witness the failure of the claim. Let

Y_{f_{0}} = {f ∈ ^{λ}2 : ∀n ∈ ω g_{f,j_{n}}(0) = g_{f_{0},j_{n}}(0)},

which by our assumption has cardinality greater than 2^{<λ}. Choose i < λ large enough so that j_{n} ∈ B(j_{m}, i) whenever n < m. For each f ∈ Y_{f_{0}}, let α(f) = min ⋂_{n<ω} C_{f,j_{n}} \ (i + 1). Define an equivalence relation E on Y_{f_{0}} by setting f_{1}Ef_{2} if and only if and for all n < ω,

• α(f_{1}) = α(f_{2});

• f_{1}↾α(f_{1}) = f_{2}↾α(f_{2}); • for all n < ω,

2Here our calculation is different from Shelah’s.

– g_{f_{1},j_{n}}↾α(f_{1}) = g_{f_{2},j_{n}}↾α(f_{2});

– C_{f_{1},j_{n}}↾α(f_{1}) = C_{f_{2},j_{n}}↾α(f_{2}).

Then E is an equivalence relation with at most λ × (2^{<λ})^{ℵ_{0}} × (2^{<λ})^{ℵ_{0}} = 2^{<λ} classes, so there are distinct functions f_{1} and f_{2} from λ to 2 such that f_{1}Ef_{2}.

Let j be the supremum of {j_{n} : n < ω}. For each ℓ ∈ {1, 2}, let C^{ℓ} = ⋂_{n<ω} C_{f_{ℓ},j_{n}} and let ⟨γ_{i}^{ℓ} : i < λ⟩ be an increasing enumeration of C^{ℓ}. For notational convenience, let γ_{λ}^{ℓ} = λ for ℓ ∈ {1, 2}.

We prove by induction on i ≤ λ that

a) γ_{i}^{1} = γ_{i}^{2};

b) for all ζ < j,

g_{f_{1},ζ}↾(γ_{i}^{1} \ {0}) = g_{f_{2},ζ}↾(γ_{i}^{2} \ {0})

and

C_{f_{1},ζ} ∩ γ_{i}^{1} = C_{f_{2},ζ} ∩ γ_{i}^{2}.

Letting i = λ and ζ = 0, this shows that f_{1} = f_{2} (since g_{f,0} = f for all f ∈ ^{λ}2), giving a contradiction and thereby proving the claim.

The induction has three cases: i = 0, i limit and i successor. For each n ∈ ω we let δ_{n} = min C_{j_{n}}^{0} = min C_{j_{n}}^{1}.

Case i = 0. For each ℓ ∈ {1, 2}, ⟨C_{f_{ℓ},j_{n}} : n < ω⟩ is a ⊆-decreasing sequence of clubs, so γ_{0}^{ℓ} = sup{δ_{n} : n < ω}. This shows (a). To see (b), fix ζ < j and n ∈ ω such that ζ < ȷ_{n} and if

g_{f_{1},ζ}↾(γ_{0}^{1} \ {0}) ≠ g_{f_{2},ζ}↾(γ_{0}^{2} \ {0})

then

g_{f_{1},ζ}↾(δ_{n} \ {0}) ≠ g_{f_{2},ζ}↾(δ_{n} \ {0})

and if

C_{f_{1},ζ} ∩ γ_{0}^{1} ≠ C_{f_{2},ζ} ∩ γ_{0}^{2}

then

C_{f_{1},ζ} ∩ δ_{n} ≠ C_{f_{2},ζ} ∩ δ_{n}.

We have assumed that g_{f_{1},j_{n}}↾δ_{n} = g_{f_{2},j_{n}}↾δ_{n}. Also, for each ℓ ∈ {1, 2}, δ_{n} ∈ C_{f_{ℓ},j_{n}}, so

F (g_{f_{ℓ},j_{n}}↾δ_{n}) = h_{f_{ℓ},j_{n}}(δ_{n}) = H(⟨α, …, C_{f_{ℓ},β} ∩ α, g_{f_{ℓ},β}↾(α \ {0}), …⟩_{β∈B_{δn}^{jn}}),

where

α = α(δ_{n}, f_{ℓ}, j_{n}) = min[ ⋂_{β∈B_{δn}^{jn}} C_{f_{ℓ},β} \ (δ_{n} + 1)].

As H is injective, it follows that

g_{f_{1},ζ}↾(δ_{n} \ {0}) = g_{f_{2},ζ}↾(δ_{n} \ {0})

and

C_{f_{1},ζ} ∩ δ_{n} = C_{f_{2},ζ} ∩ δ_{n},

which means that

g_{f_{1},ζ}↾(γ_{0}^{1} \ {0}) = g_{f_{2},ζ}↾(γ_{0}^{2} \ {0})

and

C_{f_{1},ζ} ∩ γ_{0}^{1} = C_{f_{2},ζ} ∩ γ_{0}^{2}.

This concludes the proof of the case i = 0.

The case where i is a limit ordinal is immediate.

Finally, suppose that (a) and (b) hold for i < λ, and let us see that they hold for i + 1. We have that for all n ∈ ω, g_{f_{1},j_{n}}↾γ_{0}^{1} = g_{f_{2},j_{n}}↾γ_{0}^{2}. Note that γ_{0}^{1} and γ_{0}^{2} are both nonzero, as 0 ∉ C_{f,0} for all functions f ∈ ^{λ}2. By the induction hypothesis,

g_{f_{1},j_{n}}↾(γ_{i}^{1} \ {0}) = g_{f_{2},j_{n}}↾(γ_{i}^{2} \ {0}).

Putting these two facts together, we have that

g_{f_{1},j_{n}}↾γ_{i}^{1} = g_{f_{2},j_{n}}↾γ_{i}^{2}.

For ℓ ∈ {1, 2} we have γ_{i}^{ℓ} ∈ ⋂_{n∈ω} C_{f_{ℓ},j_{n}}, so for all n ∈ ω, F (g_{f_{ℓ},j_{n}}↾γ_{i}^{ℓ}) = h_{f_{ℓ},j_{n}}(γ_{i}^{ℓ}), which is equal to

H(⟨α_{n}^{ℓ}, …, C_{f_{ℓ},β} ∩ α_{n}^{ℓ}, g_{f_{ℓ},β}↾(α_{n}^{ℓ} \ {0}), …⟩_{β∈B_{δn}^{jn}}),

where α_{n}^{ℓ} = α(γ_{i}^{ℓ}, f_{ℓ}, j_{n}) = min[⋂_{β∈B_{δn}^{jn}} C_{f_{ℓ},β} \ (γ_{i}^{ℓ} + 1)]. As H is injective, it

follows that α_{n}^{1} = α_{n}^{2} for each n ∈ ω. Since for each f ∈ ^{λ}2 and each β < β′ < λ, C_{f,β′} is contained in the limits points of C_{f,β}, it follows that ⟨α_{n}^{ℓ} : n < ω⟩ is increasing for each ℓ ∈ {1, 2}. It follows furthermore (for the same reason) that for each ℓ ∈ {1, 2},

⋃_{n∈ω} α_{n}^{ℓ} = min[ ⋂_{β<j} C_{f_{ℓ},β} \ (γ_{i}^{ℓ} + 1)] = γ_{γ+1}^{ℓ},

so γ_{i+1}^{1} = γ_{i+1}^{2} (which is part (a)). Clause (b) follows from the fact that every ζ < j is less than some j_{n}, and the fact that γ_{i+1}^{1} = γ_{i+1}^{2} is the supremum of the set of ordinals α_{n}^{1} = α_{n}^{2} for n ∈ ω.

Conclusion 0.27. If λ is regular and uncountable, μ is a cardinal such that 2^{<λ} ≤ μ < 2^{λ} and cov(μ, λ) < 2^{λ}, then Unif(λ, μ, μ, 2) fails.

The proof of Conclusion 0.27 uses Theorem 2.1(2) of [1].

Theorem 0.28. If μ > λ ≥ κ, θ = cov(μ; λ^{+}, λ^{+}, κ), cov(λ, κ, κ, 2) ≤ μ (or ≤ θ) and λ ≥ 2^{<κ} (or just θ ≥ 2^{<κ}) then cov(μ, λ^{+}, λ^{+}, 2)^{<κ} = cov(μ, λ^{+}, λ^{+}, 2).

I don’t know what the definition of cov(α, β, δ, γ) is. In PIF, Shelah says that cov(μ, λ^{+}, λ^{+}, 2) is the same as our cov(μ, λ). He also says that cov(λ, κ, κ, 2) ≤ μ follows trivially from |S_{<κ^{+}}(λ)| = λ^{κ} ≤ 2^{<λ} ≤ μ, which holds in the context of Conclusion 0.27 (I’m guessing that S_{<κ^{+}}(λ) is the set of subsets of λ of cardinality less than κ^{+}).

Proof of Conclusion 0.27. If Unif(λ, μ, μ, 2) holds, then by Lemmas 0.6(2) and 0.13, Unif(λ, μ, μ, 2^{<λ}) holds. Then by Lemma 0.19, Unif(λ, cov(μ, λ), λ, 2^{<λ}) holds, and by Lemma 0.6(4), Unif(λ, cov(μ, λ), 2^{<λ}, 2^{<λ}) holds. By Lemma 0.26, (∗)_{2^{λ},cov(μ,λ),λ^{+}} holds.

By Lemma 0.15, cov(μ, λ)^{ℵ_{0}} ≥ 2^{λ}. By Theorem 2.1(2) of [1] (and the remarks before this proof), cov(μ, λ)^{ℵ_{0}} = cov(μ, λ), which contradicts cov(μ, λ)^{ℵ_{0}} ≥ 2^{λ} > cov(μ, λ).

The assumptions for the second part of the following conclusion seem to assume that 2^{θ} ≤ μ. I don’t see why this should necessarily be true.

Conclusion 0.29. If θ < λ are regular cardinals, 2^{θ} = 2^{<λ} < 2^{λ} and μ is a cardinal such that λ ≤ μ < 2^{λ} and (∗)_{2^{λ},μ,λ^{+}} fails, then

1. Unif(λ, μ, 2^{θ}, 2^{θ}) fails;

2. if cov(μ, λ) ≤ μ (or just cov(2^{θ}, λ) ≤ μ (??)) then Unif(λ, μ, μ, λ) fails.

Proof. The first part follows immediately from Theorem 0.26, once we know that μ > λ. In any case, the failure of part one would give μ^{ℵ_{0}} ≥ 2^{λ}. If μ = λ, this would contradict the assumptions that λ has uncountable cofinality and that 2^{λ} > 2^{<λ}.

For the second part, assuming cov(μ, λ) ≤ μ, Unif(λ, μ, μ, λ) is the same as Unif(λ, μ, μ, 2^{θ}), by Lemma 0.20. (??)

I assume that the λ in the (original) first part of the following conclusion is supposed to be 2^{λ}, otherwise the first part is trivial, by the first line.

Conclusion 0.30. If θ < λ are regular cardinals, 2^{θ} = 2^{<λ} < 2^{λ} and θ ≥ ℶ_{ω} then

1. for every μ < λ, Unif(λ, μ, 2^{θ}, 2^{θ}) fails;

2. if cov(μ, λ) < 2^{λ} then Unif(λ, μ, 2^{θ}, λ) fails.

Proof. Theorem 1.10 gives us the first part, unless μ^{ℵ_{0}} ≥ 2^{λ}, so assume this to be the case. This also implies that μ > 2^{θ}. By Conclusion 0.29 then it suffices to show that (∗)_{2^{λ},μ,λ^{+}} fails. This follows from the main result of [2], which says (using μ ≥ θ ≥ 1) that μ^{[κ]} = μ for all sufficiently large regular cardinals κ ≤ ℶ_{ω}, where μ^{[κ]} is the smallest cardinality of a subset P of S_{≤κ}(μ) such that every member of S_{≤κ}(μ) is included in a union of less than κ many members of P. To see that this suffices, suppose that P is such a set for some such κ, and let P_{1} be the set of subsets of members of P of cardinality κ. Then P_{1} has cardinality at most μ × 2^{κ} ≤ μ + ℶ_{ω} = μ. If {S_{i} : i < 2^{λ}} witnesses (∗)_{2^{λ},μ,λ^{+}} we can pick for each i < 2^{λ} an a_{i} ∈ [S_{i}]^{κ}, a ζ_{i}^{∗} < κ and b_{i,ζ} ∈ P for ζ < ζ_{i}^{∗} such that a_{i} ⊆ ⋃_{ζ<ζ_{i}^{∗}} b_{i,ζ}. For each i < 2^{λ}, there is a c_{i} which is a subset of a_{i} ∩ b_{i,ζ} of cardinality κ, for some ζ < ζ_{i}^{∗}. Each c_{i} ∈ P_{i}, and as |P_{i}| ≤ μ < 2^{λ}, two distinct c_{i}’s must be the same, contradicting the assumption on {S_{i} : i < 2^{λ}}.

The second part follows from the first part and Conclusion 0.29 (2). (??)

Shelah notes that (even for smaller λ ?) it is not clear if (∗)_{2^{λ},μ,λ^{+}} is consistent with ZFC.

References

[1] S. Shelah, Further cardinal arithmetic, Israel J Math 95 (1996) 61-114

[2] S. Shelah, The Generalized Continuum Hypothesis revisited, Israel J Math 116 (2000) 285-321