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Squares, ascent paths, and chain
conditions
(Joint work with Philipp L¨
ucke)
Chris Lambie-Hanson
Department of Mathematics
Bar-Ilan University
Joint Mathematics Meetings
AMS-ASL Special Session on
Set Theory, Logic and Ramsey Theory
San Diego, CA
11 January 2018
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I. Introduction
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Special Trees
Definition
Suppose κ is a regular cardinal and (T,
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Special Trees
Definition
Suppose κ is a regular cardinal and (T,
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Special Trees
Definition
Suppose κ is a regular cardinal and (T,
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Special Trees
Definition
Suppose κ is a regular cardinal and (T,
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Special Trees
Definition
Suppose κ is a regular cardinal and (T,
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Ascent Paths
Definition
Suppose λ < κ are regular cardinals and (T,
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Ascent Paths
Definition
Suppose λ < κ are regular cardinals and (T,
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Ascent Paths
Theorem (Shelah, Todorcevic)
Suppose that λ < κ are regular cardinals, κ is not the successor
of a cardinal of cofinality λ, and (T,
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Ascent Paths
Theorem (Shelah, Todorcevic)
Suppose that λ < κ are regular cardinals, κ is not the successor
of a cardinal of cofinality λ, and (T,
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Ascent Paths
Theorem (Shelah, Todorcevic)
Suppose that λ < κ are regular cardinals, κ is not the successor
of a cardinal of cofinality λ, and (T,
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Chain Conditions
Definition
Let κ be a regular cardinal, and let P be a poset.
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Chain Conditions
Definition
Let κ be a regular cardinal, and let P be a poset.
1 P has the κ-chain condition (κ-c.c.) if every antichain of P
has size < κ.
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Chain Conditions
Definition
Let κ be a regular cardinal, and let P be a poset.
1 P has the κ-chain condition (κ-c.c.) if every antichain of P
has size < κ.
2 P is κ-Knaster if, whenever pα | α < κ is a sequence of
conditions from P, there is an unbounded A ⊆ κ such that
pα | α ∈ A consists of pairwise-compatible conditions.
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Chain Conditions
Definition
Let κ be a regular cardinal, and let P be a poset.
1 P has the κ-chain condition (κ-c.c.) if every antichain of P
has size < κ.
2 P is κ-Knaster if, whenever pα | α < κ is a sequence of
conditions from P, there is an unbounded A ⊆ κ such that
pα | α ∈ A consists of pairwise-compatible conditions.
3 P is κ-stationarily layered if the collection of regular
suborders of P is stationary in Pκ(P).
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Chain Conditions
Definition
Let κ be a regular cardinal, and let P be a poset.
1 P has the κ-chain condition (κ-c.c.) if every antichain of P
has size < κ.
2 P is κ-Knaster if, whenever pα | α < κ is a sequence of
conditions from P, there is an unbounded A ⊆ κ such that
pα | α ∈ A consists of pairwise-compatible conditions.
3 P is κ-stationarily layered if the collection of regular
suborders of P is stationary in Pκ(P).
It is not hard to show that κ-stationarily layered ⇒ κ-Knaster ⇒
κ-c.c.
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Productivity of Chain Conditions
Questions about the productivity κ-c.c. have spawned a great
deal of research in set theory. Here, we look at the productivity of
its strengthenings.
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Productivity of Chain Conditions
Questions about the productivity κ-c.c. have spawned a great
deal of research in set theory. Here, we look at the productivity of
its strengthenings.
Easily, the κ-Knaster property is always productive, i.e., if P and
Q are κ-Knaster, then P × Q is κ-Knaster.
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Productivity of Chain Conditions
Questions about the productivity κ-c.c. have spawned a great
deal of research in set theory. Here, we look at the productivity of
its strengthenings.
Easily, the κ-Knaster property is always productive, i.e., if P and
Q are κ-Knaster, then P × Q is κ-Knaster.
If κ is weakly compact, then the κ-Knaster property is
θ-productive for every θ < κ, i.e., whenever {Pα | α < θ} is a
collection of κ-Knaster posets, then the full-support product
α<θ Pα is κ-Knaster.
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Productivity of Chain Conditions
Questions about the productivity κ-c.c. have spawned a great
deal of research in set theory. Here, we look at the productivity of
its strengthenings.
Easily, the κ-Knaster property is always productive, i.e., if P and
Q are κ-Knaster, then P × Q is κ-Knaster.
If κ is weakly compact, then the κ-Knaster property is
θ-productive for every θ < κ, i.e., whenever {Pα | α < θ} is a
collection of κ-Knaster posets, then the full-support product
α<θ Pα is κ-Knaster.
Question
Under what conditions is the κ-Knaster property infinitely
productive?
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Strong Failures of Ramsey’s
Theorem
Both questions involve strong failures to higher analogues of
Ramsey’s theorem.
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Strong Failures of Ramsey’s
Theorem
Both questions involve strong failures to higher analogues of
Ramsey’s theorem.
Namely, if κ → (κ)2
λ
, then:
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Strong Failures of Ramsey’s
Theorem
Both questions involve strong failures to higher analogues of
Ramsey’s theorem.
Namely, if κ → (κ)2
λ
, then:
• Every κ-tree with a λ-ascent path has a chain of length κ.
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Strong Failures of Ramsey’s
Theorem
Both questions involve strong failures to higher analogues of
Ramsey’s theorem.
Namely, if κ → (κ)2
λ
, then:
• Every κ-tree with a λ-ascent path has a chain of length κ.
• If {Pi | i < λ} are posets and i<λ Pi is not κ-Knaster, then
one of the Pi ’s is not κ-c.c.
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II. Squares
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(κ, λ)
Definition
Suppose 0 < λ < κ are cardinals, with κ regular. (κ, λ) is the
assertion that there is a sequence Cα | α ∈ acc(κ) such that:
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(κ, λ)
Definition
Suppose 0 < λ < κ are cardinals, with κ regular. (κ, λ) is the
assertion that there is a sequence Cα | α ∈ acc(κ) such that:
1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that
0 < |Cα| ≤ λ;
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(κ, λ)
Definition
Suppose 0 < λ < κ are cardinals, with κ regular. (κ, λ) is the
assertion that there is a sequence Cα | α ∈ acc(κ) such that:
1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that
0 < |Cα| ≤ λ;
2 for all α < β in acc(κ) and all C ∈ Cβ, if α ∈ acc(C), then
C ∩ α ∈ Cα;
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(κ, λ)
Definition
Suppose 0 < λ < κ are cardinals, with κ regular. (κ, λ) is the
assertion that there is a sequence Cα | α ∈ acc(κ) such that:
1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that
0 < |Cα| ≤ λ;
2 for all α < β in acc(κ) and all C ∈ Cβ, if α ∈ acc(C), then
C ∩ α ∈ Cα;
3 there is no club D in κ such that, for all α ∈ acc(D),
D ∩ α ∈ Cα.
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(κ, λ)
Definition
Suppose 0 < λ < κ are cardinals, with κ regular. (κ, λ) is the
assertion that there is a sequence Cα | α ∈ acc(κ) such that:
1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that
0 < |Cα| ≤ λ;
2 for all α < β in acc(κ) and all C ∈ Cβ, if α ∈ acc(C), then
C ∩ α ∈ Cα;
3 there is no club D in κ such that, for all α ∈ acc(D),
D ∩ α ∈ Cα.
(κ, 1) is more commonly denoted by (κ).
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(κ, λ)
Definition
Suppose 0 < λ < κ are cardinals, with κ regular. (κ, λ) is the
assertion that there is a sequence Cα | α ∈ acc(κ) such that:
1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that
0 < |Cα| ≤ λ;
2 for all α < β in acc(κ) and all C ∈ Cβ, if α ∈ acc(C), then
C ∩ α ∈ Cα;
3 there is no club D in κ such that, for all α ∈ acc(D),
D ∩ α ∈ Cα.
(κ, 1) is more commonly denoted by (κ).
(κ, λ) can be seen as an instance of incompactness at κ.
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(κ, λ)
Definition
Suppose 0 < λ < κ are cardinals, with κ regular. (κ, λ) is the
assertion that there is a sequence Cα | α ∈ acc(κ) such that:
1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that
0 < |Cα| ≤ λ;
2 for all α < β in acc(κ) and all C ∈ Cβ, if α ∈ acc(C), then
C ∩ α ∈ Cα;
3 there is no club D in κ such that, for all α ∈ acc(D),
D ∩ α ∈ Cα.
(κ, 1) is more commonly denoted by (κ).
(κ, λ) can be seen as an instance of incompactness at κ.
If κ is a regular cardinal and (κ) fails, then κ is weakly compact
in L.
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ind(κ, λ)
Definition
Suppose that λ < κ are infinite, regular cardinals. ind(κ, λ) is
the assertion that there is a sequence
Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that:
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ind(κ, λ)
Definition
Suppose that λ < κ are infinite, regular cardinals. ind(κ, λ) is
the assertion that there is a sequence
Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that:
1 for all α ∈ acc(κ) and all i(α) ≤ i < λ, Cα,i is club in α;
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ind(κ, λ)
Definition
Suppose that λ < κ are infinite, regular cardinals. ind(κ, λ) is
the assertion that there is a sequence
Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that:
1 for all α ∈ acc(κ) and all i(α) ≤ i < λ, Cα,i is club in α;
2 for all α ∈ acc(κ) and all i(α) ≤ i < j < λ, Cα,i ⊆ Cα,j ;
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ind(κ, λ)
Definition
Suppose that λ < κ are infinite, regular cardinals. ind(κ, λ) is
the assertion that there is a sequence
Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that:
1 for all α ∈ acc(κ) and all i(α) ≤ i < λ, Cα,i is club in α;
2 for all α ∈ acc(κ) and all i(α) ≤ i < j < λ, Cα,i ⊆ Cα,j ;
3 for all α < β in acc(κ) and all i(β) ≤ i < λ, if α ∈ acc(Cβ,i ),
then i(α) ≤ i and Cβ,i ∩ α = Cα,i ;
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ind(κ, λ)
Definition
Suppose that λ < κ are infinite, regular cardinals. ind(κ, λ) is
the assertion that there is a sequence
Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that:
1 for all α ∈ acc(κ) and all i(α) ≤ i < λ, Cα,i is club in α;
2 for all α ∈ acc(κ) and all i(α) ≤ i < j < λ, Cα,i ⊆ Cα,j ;
3 for all α < β in acc(κ) and all i(β) ≤ i < λ, if α ∈ acc(Cβ,i ),
then i(α) ≤ i and Cβ,i ∩ α = Cα,i ;
4 for all α < β in acc(κ), there is i(β) ≤ i < λ such that
α ∈ acc(Cβ,i );
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ind(κ, λ)
Definition
Suppose that λ < κ are infinite, regular cardinals. ind(κ, λ) is
the assertion that there is a sequence
Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that:
1 for all α ∈ acc(κ) and all i(α) ≤ i < λ, Cα,i is club in α;
2 for all α ∈ acc(κ) and all i(α) ≤ i < j < λ, Cα,i ⊆ Cα,j ;
3 for all α < β in acc(κ) and all i(β) ≤ i < λ, if α ∈ acc(Cβ,i ),
then i(α) ≤ i and Cβ,i ∩ α = Cα,i ;
4 for all α < β in acc(κ), there is i(β) ≤ i < λ such that
α ∈ acc(Cβ,i );
5 there is no club D in κ and i < λ such that, for all
α ∈ acc(D), we have i(α) ≤ i and D ∩ α = Cα,i .
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Implications
Clearly, ind(κ, λ) ⇒ (κ, λ).
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Implications
Clearly, ind(κ, λ) ⇒ (κ, λ).
Theorem (LH-L¨
ucke)
If λ < κ are infinite, regular cardinals and (κ) holds, then
ind(κ, λ) holds.
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Implications
Clearly, ind(κ, λ) ⇒ (κ, λ).
Theorem (LH-L¨
ucke)
If λ < κ are infinite, regular cardinals and (κ) holds, then
ind(κ, λ) holds.
Theorem (LH, Shani)
Assuming the consistency of large cardinals, it is consistent that
(κ, 2) holds but ind(κ, λ) fails for all regular λ such that
λ+ < κ.
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Implications
Clearly, ind(κ, λ) ⇒ (κ, λ).
Theorem (LH-L¨
ucke)
If λ < κ are infinite, regular cardinals and (κ) holds, then
ind(κ, λ) holds.
Theorem (LH, Shani)
Assuming the consistency of large cardinals, it is consistent that
(κ, 2) holds but ind(κ, λ) fails for all regular λ such that
λ+ < κ.
Theorem (Hayut-LH)
Assuming the consistency of large cardinals, it is consistent that
ind(κ, λ) holds but (κ, θ) fails for all θ < λ.
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III. Ascent Paths
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Ascent paths from squares
Theorem (L¨
ucke)
Suppose that λ < κ are regular cardinals, S ⊆ Sκ
λ
is stationary,
and there is a (κ)-sequence Cα | α ∈ acc(κ) such that
acc(Cα) ∩ S = ∅ for all α ∈ acc(κ). Then there is a κ-Aronszajn
tree with a λ-ascent path.
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Ascent paths from squares
Theorem (L¨
ucke)
Suppose that λ < κ are regular cardinals, S ⊆ Sκ
λ
is stationary,
and there is a (κ)-sequence Cα | α ∈ acc(κ) such that
acc(Cα) ∩ S = ∅ for all α ∈ acc(κ). Then there is a κ-Aronszajn
tree with a λ-ascent path.
Theorem (LH-L¨
ucke)
Suppose that λ < κ are regular cardinals and ind(κ, λ) holds.
Then there is a κ-Aronszajn tree with a λ-ascent path.
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Ascent paths from squares
Theorem (L¨
ucke)
Suppose that λ < κ are regular cardinals, S ⊆ Sκ
λ
is stationary,
and there is a (κ)-sequence Cα | α ∈ acc(κ) such that
acc(Cα) ∩ S = ∅ for all α ∈ acc(κ). Then there is a κ-Aronszajn
tree with a λ-ascent path.
Theorem (LH-L¨
ucke)
Suppose that λ < κ are regular cardinals and ind(κ, λ) holds.
Then there is a κ-Aronszajn tree with a λ-ascent path.
Corollary (LH-L¨
ucke)
Suppose that λ < κ are regular cardinals and (κ) holds. Then
there is a κ-Aronszajn tree with a λ-ascent path.
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Tree properties
Recall that, if κ is a regular cardinal, then the tree property at κ
asserts that every κ-tree has a chain of length κ.
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Tree properties
Recall that, if κ is a regular cardinal, then the tree property at κ
asserts that every κ-tree has a chain of length κ.
For a regular λ < κ, one can formulate a similar property by
asserting that every κ-tree has a λ-ascent path.
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Tree properties
Recall that, if κ is a regular cardinal, then the tree property at κ
asserts that every κ-tree has a chain of length κ.
For a regular λ < κ, one can formulate a similar property by
asserting that every κ-tree has a λ-ascent path. Clearly, this
property is implied by the tree property; a natural question is
whether it is strictly weaker than the tree property.
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Tree properties
Recall that, if κ is a regular cardinal, then the tree property at κ
asserts that every κ-tree has a chain of length κ.
For a regular λ < κ, one can formulate a similar property by
asserting that every κ-tree has a λ-ascent path. Clearly, this
property is implied by the tree property; a natural question is
whether it is strictly weaker than the tree property.
Theorem (LH-L¨
ucke)
The following are equiconsistent over ZFC:
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Tree properties
Recall that, if κ is a regular cardinal, then the tree property at κ
asserts that every κ-tree has a chain of length κ.
For a regular λ < κ, one can formulate a similar property by
asserting that every κ-tree has a λ-ascent path. Clearly, this
property is implied by the tree property; a natural question is
whether it is strictly weaker than the tree property.
Theorem (LH-L¨
ucke)
The following are equiconsistent over ZFC:
1 there is a weakly compact cardinal;
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Tree properties
Recall that, if κ is a regular cardinal, then the tree property at κ
asserts that every κ-tree has a chain of length κ.
For a regular λ < κ, one can formulate a similar property by
asserting that every κ-tree has a λ-ascent path. Clearly, this
property is implied by the tree property; a natural question is
whether it is strictly weaker than the tree property.
Theorem (LH-L¨
ucke)
The following are equiconsistent over ZFC:
1 there is a weakly compact cardinal;
2 there are ℵ2-Aronszajn trees, but every ℵ2-Aronszajn tree
has an ℵ0-ascent path.
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IV. Chain Conditions
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Failure of Infinite Productivity
Theorem (LH-L¨
ucke)
Suppose κ is a regular, uncountable cardinal and ind(κ, ℵ0)
holds. Then there is are posets {Pi | i < ω} such that:
1 each Pi is κ-Knaster;
2 i<ω Pi is not κ-c.c.
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Failure of Infinite Productivity
Theorem (LH-L¨
ucke)
Suppose κ is a regular, uncountable cardinal and ind(κ, ℵ0)
holds. Then there is are posets {Pi | i < ω} such that:
1 each Pi is κ-Knaster;
2 i<ω Pi is not κ-c.c.
Proof Sketch.
Let Cα,i | α ∈ acc(κ), i(α) ≤ i < ω be a ind(κ, ℵ0)-sequence.
We can arrange so that, for all i < ω, {α ∈ acc(κ) | i(α) = i} is
stationary.
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Failure of Infinite Productivity
Theorem (LH-L¨
ucke)
Suppose κ is a regular, uncountable cardinal and ind(κ, ℵ0)
holds. Then there is are posets {Pi | i < ω} such that:
1 each Pi is κ-Knaster;
2 i<ω Pi is not κ-c.c.
Proof Sketch.
Let Cα,i | α ∈ acc(κ), i(α) ≤ i < ω be a ind(κ, ℵ0)-sequence.
We can arrange so that, for all i < ω, {α ∈ acc(κ) | i(α) = i} is
stationary. For each i < ω, let Ti be the tree whose underlying set
is {α ∈ acc(κ) | i(α) ≤ i}, ordered by α
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Proof Sketch (Cont.)
Let Pi be the poset consisting of finite partial functions from Ti
to ω that are injective on
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Proof Sketch (Cont.)
Let Pi be the poset consisting of finite partial functions from Ti
to ω that are injective on
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Proof Sketch (Cont.)
Let Pi be the poset consisting of finite partial functions from Ti
to ω that are injective on
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More Recent Results
Theorem (LH-L¨
ucke)
If κ is a regular cardinal and ind(κ, ℵ0) holds, then there is a
κ-Knaster poset that is not κ-stationarily layered.
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More Recent Results
Theorem (LH-L¨
ucke)
If κ is a regular cardinal and ind(κ, ℵ0) holds, then there is a
κ-Knaster poset that is not κ-stationarily layered.
Theorem (Hayut-LH)
If κ is a regular cardinal and ind(κ, ℵ0) holds, then there are
posets {Pi | i < ω} such that each Pi is κ-stationarily layered and
i<ω Pi is not κ-c.c.
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More Recent Results
Theorem (LH-L¨
ucke)
If κ is a regular cardinal and ind(κ, ℵ0) holds, then there is a
κ-Knaster poset that is not κ-stationarily layered.
Theorem (Hayut-LH)
If κ is a regular cardinal and ind(κ, ℵ0) holds, then there are
posets {Pi | i < ω} such that each Pi is κ-stationarily layered and
i<ω Pi is not κ-c.c.
Theorem (LH-Rinot)
Suppose that κ is a successor cardinal. Then there are posets
{Pi | i < ω} such that each Pi is κ-Knaster and i<ω Pi is not
κ-c.c.
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Thank you!