ANTICHAINS IN PRODUCTS OF LINEAR ORDERS

MARTIN GOLDSTERN AND SAHARON SHELAH

Abstract. We show that:

1. For many regular cardinals λ (in particular, for all successors of singular strong limit cardinals, and for all successors of singular ω-limits), for all n ∈ {2, 3, 4, … }: There is a linear order L such that L^{n} has no (incomparability-)antichain of cardinality λ, while L^{n+1} has an antichain of cardinality λ.

2. For any nondecreasing sequence (λ_{n} : n ∈ {2, 3, 4, … }) of infinite cardinals it is consistent that there is a linear order L such that, for all n: L^{n} has an antichain of cardinality λ_{n}, but no antichain of cardinality λ_{n}^{+}.

1. Introduction

Definition 1.1. For any partial ordering (P, ≤) define inc(P) as

inc(P) = sup{|A|^{+} : A ⊆ P is an antichain}

Here, an antichain is a set of pairwise incomparable elements.

In other words, κ < inc(P) iff there is an antichain of cardinality κ.

Haviar and Ploscica in [2] asked: Can there be a linear ordering L such that inc(L^{n}) ≠ inc(L^{k}) for some natural numbers n, k? (Here, L^{n} and L^{k} carry the product, or pointwise, order.)

Farley [1] has shown that for any singular cardinal κ there is a linear order L of size κ such that inc(L^{2}) = κ, inc(L^{3}) = κ^{+}.

We will be mainly interested in this question for regular cardinals. First we show in ZFC that there are many successor cardinals λ (including ℵ_{ω+1}) with the following property:

For every n ≥ 2 there is a linear order L of size λ such that inc(L^{n}) ≤ λ, inc(L^{n+1}) = λ^{+}.

This proof is given in section 2. It uses a basic fact from pcf theory.

We then show that there are (consistently) many possible behaviours for the sequence (inc(L^{n}) : n = 2, 3, 4, …). More precisely, we show that for any nondecreasing sequence of infinite cardinals (λ_{n} : 2 ≤ n < ω) there is a cardinal-preserving extension of the universe in which we can find a linear order L such that for all n ∈ {2, 3, … }: inc(L^{n}) = λ_{n}^{+}.

The first author is partially supported by the Austrian Science Foundation, FWF grant P13325-MAT.

The second author is supported by the German-Israeli Foundation for Scientific Research & Development Grant No. G-294.081.06/93. Publication number 696.

For example, it is consistent that there is a linear order L such that L^{2} has no uncountable antichain, while L^{3} does.

Here we use forcing. The heart of this second proof is the well-known Δ-system lemma.

2. A ZFC proof

Let μ be a regular cardinal. We will write D_{μ}^{bd} for the filter of cobounded sets, i.e., the filter dual to the ideal {A ⊆ μ : ∃i < μ A ⊆ i}.

∏_{i<μ} λ_{i} is the set of all functions f with domain μ satisfying f(i) < λ_{i} for all i. The relation f ∼_{D_{μ}^{bd}} g ⇔ {i : f(i) = g(i)} ∈ D_{μ}^{bd} is an equivalence relation. We call the quotient structure ∏_{i} λ_{i}/D_{μ}^{bd} (and we often do not distinguish between a function f and its equivalence class). ∏_{i} λ_{i}/D_{μ}^{bd} is partially ordered by the relation

f <_{D_{μ}^{bd}} g iff {i < μ : f(i) < g(i)} ∈ D_{μ}^{bd}

For any partial order (P, ≤) and any regular cardinal λ we say λ = tcf(P) (“λ is the true cofinality of P ”) iff there is an increasing sequence (p_{i} : i < λ) such that ∀p ∈ P ∃i < λ : p ≤ p_{i}.

Theorem 2.1. Assume that 1. (λ_{i} : i < μ) is an increasing sequence of regular cardinals

2. For each j < λ, |∏_{i<j} λ_{i}| < λ_{j}

3. λ is regular and tcf(∏ λ_{i}/D_{μ}^{bd}) = λ,

4. n ≥ 2. Then there is a linear order J of size λ such that

• J^{n+1} has an antichain of size λ

• J^{n} has no antichain of size λ

To show that there are indeed many cardinals λ which can appear as the true cofinality of a reduced product as in the assumptions of the above theorem, we use theorem II.1.5 from [3, page 50]:

Theorem 2.2. If κ is singular, cfκ = μ, then for some strictly increasing sequence ⟨λ_{i} : i < μ⟩ of regular cardinals, κ = ∑_{i<μ} λ_{i} and (∏_{i<μ} λ_{i}, <_{D_{μ}^{bd}}) has true cofinality κ^{+}.

It is clear that in the above theorem we can replace (λ_{i} : i < μ) by any increasing cofinal subsequence (λ_{f(j)} : j < μ).

Conclusion 2.3. Whenever λ = κ^{+} is the successor of a singular cardinal κ such that

1. Either κ is a strong limit cardinal

2. or at least ∀κ′ < κ : (κ′)^{<cf(κ)} < κ then we can find a sequence (λ_{i} : i < cf(κ)) as in the assumption of theorem 2.1. For example, if λ = ℵ_{ω+1}, then there is an increasing sequence (n_{k} : k ∈ ω) of natural numbers such that tcf(∏_{k∈ω} ℵ_{n_{k}}/D_{ω}^{bd}) = ℵ_{ω+1}.

Proof. Let μ = cf(κ). We can start with a sequence (λ_{i} : i < μ) such that ∏_{i<μ} λ_{i}/D_{μ}^{bd} has true cofinality λ = κ^{+}. Choose f : μ → μ increasing cofinal such that ∏ λ_{j<i}λ_{f(j)} < λ_{i} for all i, then (λ_{f(j)} : j < μ) is as required.

The proof of theorem 2.1 will occupy the rest of this section. We fix a sequence (f_{α} : α < λ) as in the assumption of the theorem. We start by writing μ = ⋃_{ℓ=0}^{n} A_{ℓ} as a disjoint union of n + 1 many D_{μ}^{bd}-positive (i.e., unbounded) sets. For ℓ = 0, …, n we define a linear order <_{ℓ} on λ as follows:

Definition 2.4. For any two distinct functions f, g ∈ ∏_{i} λ_{i} we define

d(f, g) = sup{i < μ : f↾i = g↾i} = max{i < μ : f↾i = g↾i} (1)

That is, if f ≠ g we have that d(f, g) = min{j : f(j) ≠ g(j)} is the first point where f and g diverge.

For α, β ∈ λ we define α <_{ℓ} β iff:

letting i_{α,β} := d(f_{α}, f_{β}),

either i_{α,β} ∈ A_{ℓ} and f_{α}(i_{α,β}) < f_{β}(i_{α,β})
or i_{α,β} ∉ A_{ℓ} and f_{α}(i_{α,β}) > f_{β}(i_{α,β})

(2)

We leave it to the reader to check that <_{ℓ} is indeed a linear order on λ.

We now define J to be the “ordinal sum” of all the orders <_{ℓ}:

Definition 2.5. Let

J = ⋃_{ℓ=0}^{n}{ℓ} × (λ, <_{ℓ})

with the “lexicographic” order, i.e., ⟨ℓ_{1}, α_{1}⟩ < ⟨ℓ_{2}, α_{2}⟩ iff ℓ_{1} < ℓ_{2}, or ℓ_{1} = ℓ_{2} and α_{1} <_{ℓ_{1}} α_{2}.

Claim 2.6. J^{n+1} has an antichain of size λ.

Proof. Let t⃗_{α} = ((0, α), …, (n, α)) ∈ J^{n+1}.

For any α ≠ β we have to check that t⃗_{α} and t⃗_{β} are incomparable. Let i^{∗} = d(f_{α}, f_{β}), and find ℓ^{∗} such that i^{∗} ∈ A_{ℓ^{∗}}. Wlog assume f_{α}(i^{∗}) < f_{β}(i^{∗}). Then α <_{ℓ^{∗}} β, but α >_{ℓ} β for all ℓ ≠ ℓ^{∗}, i.e., ⟨ℓ^{∗}, α⟩ <_{J} ⟨ℓ^{∗}, β⟩, but ⟨ℓ, α⟩ >_{J} ⟨ℓ, β⟩ for all ℓ ≠ ℓ^{∗}.

Proof of 2.1. It remains to show that J^{n} does not have an antichain of size λ. Towards a contradiction, assume that (t⃗_{β} : β < λ) is an antichain in J^{m}, m ≤ n, and m as small as possible. Let t⃗_{β} = (t_{β}(1), …, t_{β}(m)) ∈ J^{m}. For k = 1, …, m we can find functions ℓ_{k}, ξ_{k} such that

∀β < λ ∀k : t_{β}(k) = ⟨ℓ_{k}(β), ξ_{k}(β)⟩

Thinning out we may assume that the functions ℓ_{1}, …, ℓ_{m} are constant. We will again write ℓ_{1}, …, ℓ_{m} for those constant values.

We may also assume that for each k the function β ↦ ξ_{k}(β) is either constant or strictly increasing. If any of the functions ξ_{k} is constant we get a contradiction to the minimality of m, so all the ξ_{k} are strictly increasing. So we may moreover assume that β < γ implies ξ_{k}(β) < ξ_{k′}(γ) for all k, k′, and in particular β ≤ ξ_{k}(β) for all β, k.

Now define g_{β}^{+}, g_{β}^{−} ∈ ∏_{i<μ} λ_{i} for every β < λ as follows:

\begin{aligned}g_{β}^{+}(i) & = max(f_{ξ_{1}(β)}(i), …, f_{ξ_{n}(β)}(i)) \\ g_{β}^{−}(i) & = min(f_{ξ_{1}(β)}(i), …, f_{ξ_{n}(β)}(i)) \end{aligned} (3)

Subclaim. The set

(4) C := {i < μ : ∀β {g_{γ}^{−}(i) : γ > β} is unbounded in λ_{i}}

is in the filter D_{μ}^{bd}, i.e., its complement

S := {i < μ : ∃β < λ ∃s < λ_{i} {g_{γ}^{−}(i) : γ > β} ⊆ s}

is in the ideal dual to D_{μ}^{bd} (i.e., is a bounded set).

Proof of the subclaim. For each i ∈ S let β_{i} < λ and h(i) < λ_{i} be such that {g_{γ}^{−}(i) : γ > β_{i}} ⊆ h(i). Let β^{∗} = sup{β_{i} : i ∈ S} < λ, and extend h arbitrarily to a total function on μ. Since the sequence (f_{α} : α < λ) is cofinal in ∏_{i} λ_{i}/D_{μ}^{bd}, we can find γ > β^{∗} such that h <_{D_{μ}^{bd}} f_{γ}.

We have γ ≤ ξ_{k}(γ) for all k ∈ {1, …, m}, so the sets

X_{k} := {i < μ : h(i) < f_{ξ_{k}(γ)}(i)}

are all in D_{μ}^{bd}. Now if S were positive mod D_{μ}^{bd}, then we could find i_{∗} ∈ S ∩ X_{1} ∩ ⋯ ∩ X_{m}. But then i^{∗} ∈ X_{1} ∩ ⋯ ∩ X_{m} implies

h(i_{∗}) < g_{γ}^{−}(i_{∗}),

and i ∈ S implies

g_{γ}^{−}(i_{∗}) < h(i_{∗}),

a contradiction.

This shows that C is indeed a set in the filter D_{μ}^{bd}.

We will now use the fact that m < n + 1. Let

ℓ^{∗} ∈ {0, …, n} \ {ℓ_{1}, …, ℓ_{m}}.

Since A_{ℓ^{∗}} is positive mod D_{μ}^{bd}, we can pick

i^{∗} ∈ A_{ℓ^{∗}} ∩ C (5)

Using the fact that i^{∗} ∈ C and definition (4) we can find a sequence (β_{σ} : σ < λ_{i^{∗}}) such that

∀σ < σ′ < λ_{i^{∗}} : g_{β_{σ}}^{+}(i^{∗}) < g_{β_{σ′}}^{−} (i^{∗}), (6)

We now restrict our attention from (t⃗_{β} : β < λ) to the subsequence (t⃗_{β_{σ}} : σ < λ_{i^{∗}}); we will show that this sequence cannot be an antichain. For notational simplicity only we will assume β_{σ} = σ for all σ < λ_{i^{∗}}.

Recall that t⃗_{σ} = (⟨ℓ_{1}, ξ_{1}(σ)⟩, …, ⟨ℓ_{m}, ξ_{m}(σ)⟩). For each σ < λ_{i^{∗}} define

x⃗_{σ} := (f_{ξ_{1}(σ)}↾i^{∗}, …, f_{ξ_{n}(σ)}↾i^{∗}) ∈ (∏_{j<i^{∗}} λ_{j})^{m}. Since |∏_{j<i^{∗}} λ_{j}| < λ_{i^{∗}},

there are only < λ_{i^{∗}} many possible values for x⃗_{σ}, so we can find σ_{1} < σ_{2} < λ_{i^{∗}} such that x⃗_{σ_{1}} = x⃗_{σ_{2}}.

Now note that by (3) and (6) we have

f_{ξ_{k}(σ_{1})}(i^{∗}) ≤ g_{σ_{1}}^{+}(i^{∗}) < g_{σ_{2}}^{−}(i^{∗}) ≤ f_{ξ_{k}(σ_{2})}(i^{∗}). (7)

Hence d(f_{ξ_{k}(σ_{1})}, f_{ξ_{k}(σ_{2})}) = i^{∗} for k = 1, …, m.

Since i^{∗} ∈ A_{ℓ^{∗}} we have for all k: i^{∗} ∉ A_{ℓ_{k}} . From (1), (2), (7) we get

ξ_{k}(σ_{1}) <_{ℓ_{k}} ξ_{k}(σ_{2}) for k = 1, …, m.

Hence ⟨ℓ_{k}, ξ_{k}(σ_{1})⟩ < ⟨ℓ_{k}, ξ_{k}(σ_{2})⟩ for all k, which means t⃗_{σ_{1}} < t⃗_{σ_{2}}.

3. Consistency

Theorem 3.1. Assume ℵ_{0} ≤ λ_{2} ≤ λ_{3} ≤ ⋯, λ_{n}^{<κ} = λ_{n}. κ^{<κ} = κ. Then there is a forcing notion ℙ which satisfies the κ-cc and is κ-complete, and a ℙ-name \begin{matrix}I \\ \end{matrix}such that

⊩_{P} \begin{matrix}I \\ \end{matrix}⊆ 2^{κ}, inc(\begin{matrix}I \\ \end{matrix}^{n}) = λ_{n}^{+}

Remark 3.2. At first reading, the reader may want to consider the special case κ = ℵ_{0}, λ_{n+2} = ℵ_{n}.

Notation 3.3. 1. We let λ_{1} = 0, λ_{ω} = sup{λ_{n} : n < ω}.

2. It is understood that 2^{α} is linearly ordered lexicographically, and (2^{α})^{m} is partially ordered by the pointwise order.

3. For α ≤ β ≤ κ, η ∈ 2^{α}, ν ∈ 2^{β}, we define

η ⊴ ν iff ν extends η, i.e., η ⊆ ν

4. For η̄ ∈ (2^{α})^{n}, η̄ = (η(0), …, η(n−1)), ν̄ ∈ (2^{β})^{n}, ν = (ν(0), …, ν(n− 1)), we let

η̄ ⊴ ν̄ iff η(0) ⊴ ν(0), …, η(n − 1) ⊴ ν(n − 1).

5. For η ∈ 2^{α}, i ∈ {0, 1} we write η^{⌢}i for the element ν ∈ 2^{α+1} satisfying η ⊴ ν, ν(α) = i.

Definition 3.4. Let η̄ ∈ (2^{α})^{m}, k ∈ {0, …, m − 1}, m ≥ 2. We define η̄^{⌢}¯1, η̄^{⌢}¯0, η̄^{⌢}{k↦1 else ¯0}, η̄^{⌢}{k↦0 else ¯1} in (2^{α+1})^{m} as follows: All four are ⊴-extensions of η̄, and:

– η̄^{⌢}¯0(n) = η(n)^{⌢}0 for all n < m.

– η̄^{⌢}¯1(n) = η(n)^{⌢}1 for all n < m.

– η̄^{⌢}{k↦0 else ¯1}(n) = η(n)^{⌢}1 for all n ≠ k, η̄^{⌢}{k↦0 else ¯1}(k) = η(n)^{⌢}0.

– η̄^{⌢}{k↦1 else ¯0}(n) = η(n)^{⌢}0 for all n ≠ k, η̄^{⌢}{k↦1 else ¯0}(k) = η(n)^{⌢}1.

0 0 0 0  1 0 0 0 k^{η̄^{⌢}¯0}  ^{η̄^{⌢}{k↦1 else ¯0}} η̄

Fact 3.5. 1. If α ≤ β ≤ κ, η̄, η̄′ ∈ (2^{α})^{n} are incomparable, ν̄, ν̄′ ∈ (2^{β})^{n},

η̄ ⊴ ν̄, η̄′ ⊴ ν̄′, then also ν̄ and ν̄′ are incomparable.

2. η̄^{⌢}¯0 < η̄^{⌢}¯1.

3. η̄^{⌢}{k↦0 else ¯1} and η̄^{⌢}{k↦1 else ¯0} are incomparable.

Definition 3.6. We let ℙ be the set of all “conditions”

p = (u^{p}, α^{p}, (η̄_{ξ}^{p} : ξ ∈ u^{p}))

satisfying the following requirements for all m:

– u^{p} ∈ [λ_{ω}]^{<κ}

– α^{p} < κ

– For all ξ ∈ u^{p} ∩ (λ_{m} \ λ_{m−1}): η̄_{ξ}^{p} = (η_{ξ}^{p}(0), …, η_{ξ}^{p}(m − 1)) ∈ (2^{α^{p}})^{m}.

– For all ξ ≠ ξ′ in u^{p} ∩ (λ_{m} \ λ_{m−1}), η̄_{ξ}^{p} and η̄_{ξ′}^{p} are incomparable in (2^{α^{p}})^{m}.

We define p ≤ q (“q is stronger than p”) iff

– u^{p} ⊆ u^{q}

– α^{p} ≤ α^{q}

– for all ξ ∈ u^{p}, η̄_{ξ}^{p} ⊴ η̄_{ξ}^{q}

Fact 3.7. 1. For all α < κ: The set {p ∈ ℙ : α^{p} ≥ α} is dense in ℙ. 2. For all ξ < λ_{ω}: The set {p ∈ ℙ : ξ ∈ u^{p}} is dense in ℙ.

Fact and Definition 3.8. We let (ν̄ ; _{ξ} : ξ < λ_{ω}) be the “generic object”, i.e., a name satisfying

∀m ∈ ω ∀p ∈ ℙ ∀ξ ∈ u^{p} ∩ (λ_{m} \ λ_{m−1}) : p ⊩_{ℙ} ν̄ ; _{ξ} ∈ (2^{κ})^{m}
∀p ∈ ℙ ∀ξ ∈ u^{p} : p ⊩ η̄_{ξ}^{p} ⊴ ν̄ ; _{ξ}

(This definition makes sense, by fact 3.7.)

Clearly, ⊩ ξ, ξ′ ∈ λ_{m} \ λ_{m−1} ⇒ ν̄ ; _{ξ}, ν̄ ; _{ξ′} incompatible.

We let ⊩ \begin{matrix}I \\ \end{matrix}= ⋃_{m=2}^{∞}\{ν_{ξ}(ℓ) : ξ ∈ λ_{m} \ λ_{m−1}, ℓ < m\}.

Theorem 3.9. Let ℙ, \begin{matrix}I \\ \end{matrix}be as in 3.6 and 3.8.

Then ⊩_{ℙ} inc(\begin{matrix}I \\ \end{matrix}^{m}) = λ_{m}.

It is clear that ℙ is κ-complete, and κ^{+}-cc is proved by an argument similar to the Δ-system argument below. So all the λ_{m} stay cardinals.

We can show by induction that ⊩ α(\begin{matrix}I \\ \end{matrix}^{m}) > λ_{m}, i.e., \begin{matrix}I \\ \end{matrix}^{m} has an antichain of size λ_{m}: This is clear if λ_{m} = λ_{m−1} (and void if m = 0); if λ_{m} > λ_{m−1} then (ν̄_{ξ} : ξ ∈ λ_{m} \ λ_{m−1}) will be forced to be antichain.

It remains to show that (for any m) there is no antichain of size λ_{m}^{+} in

\begin{matrix}I \\ \end{matrix}^{m}.

Fix m^{∗} ∈ ω, and assume wlog that λ_{m^{∗}+1} > λ_{m^{∗}}.

[Why is this no loss of generality? If λ_{m^{∗}} = λ_{ω}, then the cardinality of \begin{matrix}I \\ \end{matrix} is at most λ_{m^{∗}}, and there is nothing to prove. If λ_{m^{∗}} = λ_{m^{∗}+1} < λ_{ω}, then replace m^{∗} by min{m ≥ m^{∗} : λ_{m} < λ_{m+1}}]

Towards a contradiction, assume that there is a condition p and a sequence of names ⟨ρ̄ ; _{β} : β < λ_{m^{∗}}^{+}⟩ such that

p ⊩ ⟨ρ̄ ; _{β} : β < λ_{m^{∗}}^{+}⟩ is an antichain in \begin{matrix}I \\ \end{matrix}^{m^{∗}}

Let ρ̄ ; _{β} = (\begin{matrix}ρ \\ \end{matrix}_{β}(n) : n < m^{∗}). For each β < λ_{m^{∗}}^{+} and each n < m^{∗} we can find a condition p_{β} ≥ p and

m(β, n) ∈ ω  ℓ(β, n) < m(β, n)  ξ_{n}(β) ∈ λ_{m(β,n)} \ λ_{m(β,n)−1}

such that

p_{β} ⊩ ρ_{β}(n) = ν_{ξ_{n}(β)}(ℓ(β, n))

We will now employ a Δ-system argument. We define a family (ζ^{β} : β < λ_{m^{∗}}^{+}) of functions as follows: Let i_{β} be the

order type of u^{p_{β}}, and let

u^{β} = u^{p_{β}} = {ζ^{β}(i) : i < i_{β}} in increasing enumeration

By 3.7.2 may assume ξ_{n}(β) ∈ u^{β}, say ξ_{n}(β) = ζ^{β}(i(β, n)). By thinning out our alleged antichain ⟨ρ̄ ; _{β} : β < λ_{m^{∗}}^{+}⟩ we may assume • For some i^{∗} < κ, for all β: i_{β} = i^{∗}

• For some α^{∗} < κ, for all β: α^{p_{β}} = α^{∗}

• For each i < i^{∗} there is some m_{⟨i⟩} such that for all β: ζ^{β}(i) ∈ λ_{m_{⟨i⟩}} \ λ_{m_{⟨i⟩}−1}

• For each i < i^{∗} there is some η̄_{⟨i⟩} ∈ (2^{α^{∗}})^{m_{⟨i⟩}} such that for all β: η̄_{ζ^{β}(i)}^{p_{β}} = η̄_{⟨i⟩}. (Here we use λ_{m}^{<κ} = λ_{m}.)

• the family ⟨u^{β} : β < λ_{m^{∗}}^{+}⟩ is a Δ-system, i.e., there is some set u^{∗} such that for all β ≠ γ: u^{β} ∩ u^{γ} = u^{∗}.

• Moreover: there is a set Δ ⊆ i^{∗} such that for all β: u^{∗} = {ζ^{β}(i) : i ∈ Δ}. Since ζ^{β} is increasing, this also implies ζ^{β}(i) = ζ^{γ}(i) for i ∈ Δ.)

• The functions (β, n) ↦ ℓ(β, n), and (β, n) ↦ i(β, n) do not depend on β, i.e. there are (ℓ_{n} : n < m^{∗}) and (i_{n} : n < m^{∗}) such that i(β, n) = i_{n}, ℓ(β, n) = ℓ_{n} for all β.

Note that for i ∈ i^{∗} \ Δ all the ζ^{β}(i) are distinct elements of λ_{m_{⟨i⟩}}, hence:

i ∉ Δ implies λ_{m^{∗}}^{+} ≤ λ_{m_{⟨i⟩}}, hence m_{⟨i⟩} > m^{∗}. Now pick k^{∗} ≤ m^{∗} such that k^{∗} ∉ {ℓ_{n} : n < m^{∗}}. Pick any distinct

β, γ < λ_{m^{∗}}^{+}. We will find a condition q extending p_{β} and p_{γ}, such that q ⊩ ρ̄_{β} ≤ ρ̄_{γ}.

We define q as follows:

• u^{q} := u_{β} ∪ u_{γ} = u^{∗} ˙∪ {ζ^{β}(i) : i ∈ i^{∗} \ Δ} ˙∪ {ζ^{γ}(i) : i ∈ i^{∗} \ Δ}.

• α^{q} = α^{∗} + 1.

• For ξ ∈ u^{∗}, say ξ = ζ^{β}(i) = ζ^{γ}(i), recall that η̄_{ξ}^{p_{β}} = η̄_{⟨i⟩} = η̄_{ξ}^{p_{γ}}. We let η_{ξ}^{q} = η_{⟨i⟩}^{⌢}¯0 (see 3.3).

• For ξ = ζ^{β}(i), i ∈ i^{∗} \ Δ, we have η̄_{ξ}^{p_{β}} = η̄_{⟨i⟩} ∈ (2^{α^{∗}})^{m_{⟨i⟩}}, where m_{⟨i⟩} > m^{∗}. Hence η̄_{⟨i⟩}^{⌢}{ℓ↦1 else ¯0} is well-defined. We let

η̄_{ξ}^{q} = η̄_{⟨i⟩}^{⌢}{ℓ^{∗}↦1 else ¯0}

• For ξ = ζ^{γ}(i), i ∈ i^{∗} \ Δ, we let

η̄_{ξ}^{q} = η̄_{⟨i⟩}^{⌢}{k^{∗}↦0 else ¯1}

We claim that q is a condition. The only nontrivial requirement is the incompatibility of all η̄_{ξ}^{q}: Let ξ, ξ′ ∈ u^{q}, ξ ≠ ξ′.

If ξ, ξ′ ∈ u^{β}, then the incompatibility of η̄_{ξ}^{q} and η̄_{ξ′}^{q} follows from the incompatibility of η̄_{ξ}^{p_{β}} and η̄_{ξ′}^{p_{β}}. The same argument works for ξ, ξ′ ∈ u_{γ}.

So let ξ ∈ u_{β} \ u^{∗}, ξ′ ∈ u_{γ} \ u^{∗}. Say ξ = ζ^{β}(i), ξ′ = ζ^{γ}(i′). If i ≠ i′, then η̄_{⟨i⟩} = η̄_{ζ^{β}(i)}^{p_{β}} = η̄_{ζ^{γ}(i)}^{p_{γ}} and η̄_{⟨i′⟩} = η̄_{ζ^{γ}(i′)}^{p_{γ}} are incompati-

ble. From η̄_{⟨i⟩} ⊴ η̄_{ξ}^{q} and η̄_{⟨i′⟩} ⊴ η̄_{ξ′}^{q} we conclude that also η̄_{ξ}^{q} and η̄_{ξ′}^{q} are incompatible.

Finally, we consider the case i = i′. We have

η̄_{ξ}^{q} = η̄_{⟨i⟩}^{⌢}{k^{∗}↦0 else ¯1} η̄_{ξ′}^{q} = η̄_{⟨i⟩}^{⌢}{k^{∗}↦1 else ¯0}

so by 3.7.3, η̄_{ξ}^{q} and η̄_{ξ′}^{q} are incompatible.

This concludes the construction of q. We now check that q ⊩ ρ̄_{β} ≤ ρ̄_{γ}, i.e., q ⊩ ρ_{β}(n) ≤ ρ_{γ}(n) for all n. Clearly, q ⊩ ρ_{β}(n) = ν_{ζ^{β}(i_{n})}(ℓ_{n}) ⊵ η̄_{ζ^{β}(i_{n})}^{q} = η_{⟨i_{n}⟩}^{⌢}¯0. Here we use the fact that k^{∗} ≠ ℓ_{n}. Similarly, q ⊩ ρ_{γ}(n) = ν_{ζ^{β}(i_{n})}(ℓ_{n}) ⊵ η_{⟨i_{n}⟩}^{⌢}¯1.

Hence q ⊩ ρ̄_{β} ≤ ρ̄_{γ}. This concludes the proof of theorem 3.1

References

[1] Jonathan Farley. Cardinalities of infinite antichains in products of chains. Algebra Universalis, 42:235–238, 1999.

[2] M. Ploščica and M. Haviar. On order-polynomial completeness of lattices. Algebra Universalis, 39:217–219, 1998.

[3] Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Oxford University Press, 1994.

Institut für Algebra, Technische Universität Wien, Wiedner Hauptstraße 8-10/118.2, A-1040 Wien, Austria

E-mail address: Martin.Goldstern@tuwien.ac.at
URL: http://www.tuwien.ac.at/goldstern/

Department of Mathematics, Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel

E-mail address: shelah@math.huji.ac.il
URL: http://math.rutgers.edu/~shelah/