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Unbounded colorings and the
C-sequence number
(Joint work with Assaf Rinot)
Chris Lambie-Hanson
Department of Mathematics
Bar-Ilan University
SETTOP
Novi Sad, Serbia
2 July 2018
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Table of contents
∅. Three properties of weakly compact cardinals
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Table of contents
∅. Three properties of weakly compact cardinals
I. Introducing the C-sequence number
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Table of contents
∅. Three properties of weakly compact cardinals
I. Introducing the C-sequence number
II. Unbounded pair-colorings
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Table of contents
∅. Three properties of weakly compact cardinals
I. Introducing the C-sequence number
II. Unbounded pair-colorings
III. Infinite productivity of strong chain conditions
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Table of contents
∅. Three properties of weakly compact cardinals
I. Introducing the C-sequence number
II. Unbounded pair-colorings
III. Infinite productivity of strong chain conditions
IV. Finding connections
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∅
∅
∅. Weakly compact cardinals
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WCCs and C-sequences
Definition (Todorcevic)
Suppose that κ is a regular, uncountable cardinal. A C-sequence
(over κ) is a sequence Cα | α < κ such that, for all limit α < κ,
Cα is a club in α.
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WCCs and C-sequences
Definition (Todorcevic)
Suppose that κ is a regular, uncountable cardinal. A C-sequence
(over κ) is a sequence Cα | α < κ such that, for all limit α < κ,
Cα is a club in α. The C-sequence is trivial if there is an
unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ
such that ∆ ∩ α ⊆ Cβ.
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WCCs and C-sequences
Definition (Todorcevic)
Suppose that κ is a regular, uncountable cardinal. A C-sequence
(over κ) is a sequence Cα | α < κ such that, for all limit α < κ,
Cα is a club in α. The C-sequence is trivial if there is an
unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ
such that ∆ ∩ α ⊆ Cβ.
Theorem (Todorcevic)
Suppose that κ is a regular, uncountable cardinal. TFAE:
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WCCs and C-sequences
Definition (Todorcevic)
Suppose that κ is a regular, uncountable cardinal. A C-sequence
(over κ) is a sequence Cα | α < κ such that, for all limit α < κ,
Cα is a club in α. The C-sequence is trivial if there is an
unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ
such that ∆ ∩ α ⊆ Cβ.
Theorem (Todorcevic)
Suppose that κ is a regular, uncountable cardinal. TFAE:
1 κ is weakly compact;
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WCCs and C-sequences
Definition (Todorcevic)
Suppose that κ is a regular, uncountable cardinal. A C-sequence
(over κ) is a sequence Cα | α < κ such that, for all limit α < κ,
Cα is a club in α. The C-sequence is trivial if there is an
unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ
such that ∆ ∩ α ⊆ Cβ.
Theorem (Todorcevic)
Suppose that κ is a regular, uncountable cardinal. TFAE:
1 κ is weakly compact;
2 for every C-sequence Cα | α < κ , there is an unbounded
∆ ⊆ κ such that, for every α < κ, there is β < κ such that
∆ ∩ α = Cβ ∩ α.
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WCCs and pair-colorings
Definition
Suppose that θ ≤ κ are cardinals. Then κ → (κ)2
θ
is the assertion
that,
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WCCs and pair-colorings
Definition
Suppose that θ ≤ κ are cardinals. Then κ → (κ)2
θ
is the assertion
that, for every c : [κ]2 → θ,
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WCCs and pair-colorings
Definition
Suppose that θ ≤ κ are cardinals. Then κ → (κ)2
θ
is the assertion
that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such
that |c“[X]2| = 1.
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WCCs and pair-colorings
Definition
Suppose that θ ≤ κ are cardinals. Then κ → (κ)2
θ
is the assertion
that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such
that |c“[X]2| = 1.
By Ramsey’s theorem, ℵ0 → (ℵ0)2
n
for all n < ω.
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WCCs and pair-colorings
Definition
Suppose that θ ≤ κ are cardinals. Then κ → (κ)2
θ
is the assertion
that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such
that |c“[X]2| = 1.
By Ramsey’s theorem, ℵ0 → (ℵ0)2
n
for all n < ω.
For regular, uncountable κ, the following are equivalent:
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WCCs and pair-colorings
Definition
Suppose that θ ≤ κ are cardinals. Then κ → (κ)2
θ
is the assertion
that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such
that |c“[X]2| = 1.
By Ramsey’s theorem, ℵ0 → (ℵ0)2
n
for all n < ω.
For regular, uncountable κ, the following are equivalent:
1 κ is weakly compact;
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WCCs and pair-colorings
Definition
Suppose that θ ≤ κ are cardinals. Then κ → (κ)2
θ
is the assertion
that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such
that |c“[X]2| = 1.
By Ramsey’s theorem, ℵ0 → (ℵ0)2
n
for all n < ω.
For regular, uncountable κ, the following are equivalent:
1 κ is weakly compact;
2 κ → (κ)2
2
;
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WCCs and pair-colorings
Definition
Suppose that θ ≤ κ are cardinals. Then κ → (κ)2
θ
is the assertion
that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such
that |c“[X]2| = 1.
By Ramsey’s theorem, ℵ0 → (ℵ0)2
n
for all n < ω.
For regular, uncountable κ, the following are equivalent:
1 κ is weakly compact;
2 κ → (κ)2
2
;
3 κ → (κ)2
θ
for all θ < κ.
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WCCs and chain conditions
Definition
For a property P and a cardinal θ, let us say that P is
θ-productive if, whenever (Qi )i<θ all have P, then i<θ Qi also
has P. (All products here are taken with full support.)
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WCCs and chain conditions
Definition
For a property P and a cardinal θ, let us say that P is
θ-productive if, whenever (Qi )i<θ all have P, then i<θ Qi also
has P. (All products here are taken with full support.)
Definition
Suppose that κ is a regular, uncountable cardinal and Q is a
poset. Q is κ-Knaster if, whenever A ⊆ Q has size κ, there is
B ⊆ A, also of size κ, consisting of pairwise compatible
conditions.
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WCCs and chain conditions
Definition
For a property P and a cardinal θ, let us say that P is
θ-productive if, whenever (Qi )i<θ all have P, then i<θ Qi also
has P. (All products here are taken with full support.)
Definition
Suppose that κ is a regular, uncountable cardinal and Q is a
poset. Q is κ-Knaster if, whenever A ⊆ Q has size κ, there is
B ⊆ A, also of size κ, consisting of pairwise compatible
conditions.
If κ is weakly compact, then κ-Knaster = κ-c.c., and both
properties are θ-productive for all θ < κ.
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I. The C-sequence number
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The C-sequence number
Recall: κ is weakly compact iff for every C-sequence
Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every
α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α.
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The C-sequence number
Recall: κ is weakly compact iff for every C-sequence
Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every
α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α.
Definition
Suppose that κ is a regular, uncountable cardinal. The
C-sequence number, χ(κ), is defined as follows.
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The C-sequence number
Recall: κ is weakly compact iff for every C-sequence
Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every
α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α.
Definition
Suppose that κ is a regular, uncountable cardinal. The
C-sequence number, χ(κ), is defined as follows.
• If κ is weakly compact, then χ(κ) = 0.
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The C-sequence number
Recall: κ is weakly compact iff for every C-sequence
Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every
α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α.
Definition
Suppose that κ is a regular, uncountable cardinal. The
C-sequence number, χ(κ), is defined as follows.
• If κ is weakly compact, then χ(κ) = 0.
• Otherwise, χ(κ) is the smallest χ such that,
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The C-sequence number
Recall: κ is weakly compact iff for every C-sequence
Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every
α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α.
Definition
Suppose that κ is a regular, uncountable cardinal. The
C-sequence number, χ(κ), is defined as follows.
• If κ is weakly compact, then χ(κ) = 0.
• Otherwise, χ(κ) is the smallest χ such that, for every
C-sequence Cα | α < κ ,
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The C-sequence number
Recall: κ is weakly compact iff for every C-sequence
Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every
α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α.
Definition
Suppose that κ is a regular, uncountable cardinal. The
C-sequence number, χ(κ), is defined as follows.
• If κ is weakly compact, then χ(κ) = 0.
• Otherwise, χ(κ) is the smallest χ such that, for every
C-sequence Cα | α < κ , there are an unbounded ∆ ⊆ κ
and a function b : κ → [κ]χ such that,
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The C-sequence number
Recall: κ is weakly compact iff for every C-sequence
Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every
α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α.
Definition
Suppose that κ is a regular, uncountable cardinal. The
C-sequence number, χ(κ), is defined as follows.
• If κ is weakly compact, then χ(κ) = 0.
• Otherwise, χ(κ) is the smallest χ such that, for every
C-sequence Cα | α < κ , there are an unbounded ∆ ⊆ κ
and a function b : κ → [κ]χ such that, for all α < κ,
∆ ∩ α ⊆ β∈b(α)
Cβ.
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The C-sequence number
Recall: κ is weakly compact iff for every C-sequence
Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every
α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α.
Definition
Suppose that κ is a regular, uncountable cardinal. The
C-sequence number, χ(κ), is defined as follows.
• If κ is weakly compact, then χ(κ) = 0.
• Otherwise, χ(κ) is the smallest χ such that, for every
C-sequence Cα | α < κ , there are an unbounded ∆ ⊆ κ
and a function b : κ → [κ]χ such that, for all α < κ,
∆ ∩ α ⊆ β∈b(α)
Cβ.
Note that χ(κ) ∈ {0, 1} iff every C-sequence over κ is trivial.
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Some basic facts
χ(κ) can be interpreted as a sort of measure of how far away κ is
from being weakly compact. This view is bolstered by the
following facts.
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Some basic facts
χ(κ) can be interpreted as a sort of measure of how far away κ is
from being weakly compact. This view is bolstered by the
following facts. (Note that χ(κ) ≤ sup(κ ∩ Reg) for all κ.)
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Some basic facts
χ(κ) can be interpreted as a sort of measure of how far away κ is
from being weakly compact. This view is bolstered by the
following facts. (Note that χ(κ) ≤ sup(κ ∩ Reg) for all κ.)
• Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
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Some basic facts
χ(κ) can be interpreted as a sort of measure of how far away κ is
from being weakly compact. This view is bolstered by the
following facts. (Note that χ(κ) ≤ sup(κ ∩ Reg) for all κ.)
• Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
• If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0.
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Some basic facts
χ(κ) can be interpreted as a sort of measure of how far away κ is
from being weakly compact. This view is bolstered by the
following facts. (Note that χ(κ) ≤ sup(κ ∩ Reg) for all κ.)
• Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
• If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0.
• (If χ(κ) > 1, then χ(κ) ≥ ℵ0.)
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Some basic facts
χ(κ) can be interpreted as a sort of measure of how far away κ is
from being weakly compact. This view is bolstered by the
following facts. (Note that χ(κ) ≤ sup(κ ∩ Reg) for all κ.)
• Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
• If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0.
• (If χ(κ) > 1, then χ(κ) ≥ ℵ0.)
• If (κ, < ℵ0) holds, then χ(κ) = sup(κ ∩ Reg).
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Inaccessible cardinals
Theorem (LH-Rinot)
Suppose that χ ≤ κ are cardinals such that κ is weakly compact
and χ is either 1 or an infinite, regular cardinal. Then there is a
forcing extension in which κ remains inaccessible and χ(κ) = χ.
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Inaccessible cardinals
Theorem (LH-Rinot)
Suppose that χ ≤ κ are cardinals such that κ is weakly compact
and χ is either 1 or an infinite, regular cardinal. Then there is a
forcing extension in which κ remains inaccessible and χ(κ) = χ.
Sketch of proof.
First, force with an Easton-support iteration P that adds a Cohen
subset to every inaccessible α < κ. In V P, the weak compactness
of κ is preserved by adding a Cohen subset to κ.
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Inaccessible cardinals
Theorem (LH-Rinot)
Suppose that χ ≤ κ are cardinals such that κ is weakly compact
and χ is either 1 or an infinite, regular cardinal. Then there is a
forcing extension in which κ remains inaccessible and χ(κ) = χ.
Sketch of proof.
First, force with an Easton-support iteration P that adds a Cohen
subset to every inaccessible α < κ. In V P, the weak compactness
of κ is preserved by adding a Cohen subset to κ. Define Q as
follows, depending on the value of χ.
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Inaccessible cardinals
Theorem (LH-Rinot)
Suppose that χ ≤ κ are cardinals such that κ is weakly compact
and χ is either 1 or an infinite, regular cardinal. Then there is a
forcing extension in which κ remains inaccessible and χ(κ) = χ.
Sketch of proof.
First, force with an Easton-support iteration P that adds a Cohen
subset to every inaccessible α < κ. In V P, the weak compactness
of κ is preserved by adding a Cohen subset to κ. Define Q as
follows, depending on the value of χ.
• If χ = 1, let Q be Kunen’s forcing to add a homogeneous
κ-Souslin tree.
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Inaccessible cardinals
Sketch of proof, cont.
• If ℵ0 ≤ χ < κ, let Q be the forcing to add a
ind(κ, χ)-sequence.
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Inaccessible cardinals
Sketch of proof, cont.
• If ℵ0 ≤ χ < κ, let Q be the forcing to add a
ind(κ, χ)-sequence.
• If χ = κ, let Q be the forcing to add a non-reflecting
stationary subset of κ.
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Inaccessible cardinals
Sketch of proof, cont.
• If ℵ0 ≤ χ < κ, let Q be the forcing to add a
ind(κ, χ)-sequence.
• If χ = κ, let Q be the forcing to add a non-reflecting
stationary subset of κ.
V P∗ ˙
Q is the desired model. The key point is that, in all cases, the
weak compactness of κ can be resurrected by a κ-distributive
forcing.
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Inaccessible cardinals
Sketch of proof, cont.
• If ℵ0 ≤ χ < κ, let Q be the forcing to add a
ind(κ, χ)-sequence.
• If χ = κ, let Q be the forcing to add a non-reflecting
stationary subset of κ.
V P∗ ˙
Q is the desired model. The key point is that, in all cases, the
weak compactness of κ can be resurrected by a κ-distributive
forcing. If χ = 1, this forcing is, moreover, κ-cc, which
immediately gives χ(κ) = 1.
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Inaccessible cardinals
Sketch of proof, cont.
• If ℵ0 ≤ χ < κ, let Q be the forcing to add a
ind(κ, χ)-sequence.
• If χ = κ, let Q be the forcing to add a non-reflecting
stationary subset of κ.
V P∗ ˙
Q is the desired model. The key point is that, in all cases, the
weak compactness of κ can be resurrected by a κ-distributive
forcing. If χ = 1, this forcing is, moreover, κ-cc, which
immediately gives χ(κ) = 1. If χ = κ, χ(κ) = κ follows from the
fact that every stationary subset of κ ∩ cof(> χ(κ)) reflects.
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Inaccessible cardinals
Sketch of proof, cont.
• If ℵ0 ≤ χ < κ, let Q be the forcing to add a
ind(κ, χ)-sequence.
• If χ = κ, let Q be the forcing to add a non-reflecting
stationary subset of κ.
V P∗ ˙
Q is the desired model. The key point is that, in all cases, the
weak compactness of κ can be resurrected by a κ-distributive
forcing. If χ = 1, this forcing is, moreover, κ-cc, which
immediately gives χ(κ) = 1. If χ = κ, χ(κ) = κ follows from the
fact that every stationary subset of κ ∩ cof(> χ(κ)) reflects. If
ℵ0 ≤ χ < κ, a slightly more delicate argument is needed.
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Successor cardinals
If λ is an infinite cardinal, then, χ(λ+) ≤ λ.
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Successor cardinals
If λ is an infinite cardinal, then, χ(λ+) ≤ λ.
Moreover, since there is a C-sequence Cα | α < λ+ such that
otp(Cα) ≤ λ for all α < λ+, an easy ordinal arithmetic argument
yields cf(λ) ≤ χ(λ+).
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Successor cardinals
If λ is an infinite cardinal, then, χ(λ+) ≤ λ.
Moreover, since there is a C-sequence Cα | α < λ+ such that
otp(Cα) ≤ λ for all α < λ+, an easy ordinal arithmetic argument
yields cf(λ) ≤ χ(λ+).
In particular, if λ is regular, then χ(λ+) = λ.
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Successor cardinals
If λ is an infinite cardinal, then, χ(λ+) ≤ λ.
Moreover, since there is a C-sequence Cα | α < λ+ such that
otp(Cα) ≤ λ for all α < λ+, an easy ordinal arithmetic argument
yields cf(λ) ≤ χ(λ+).
In particular, if λ is regular, then χ(λ+) = λ.
If λ is a singular cardinal, the story is more interesting.
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Successors of singular cardinals
Theorem (LH-Rinot)
Suppose that λ is a singular limit of supercompact cardinals.
Then χ(λ+) = cf(λ).
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Successors of singular cardinals
Theorem (LH-Rinot)
Suppose that λ is a singular limit of supercompact cardinals.
Then χ(λ+) = cf(λ).
Moreover, if θ ∈ (cf(λ), λ) is a regular cardinal, then there is a
cofinality-preserving forcing extension in which χ(λ+) = θ.
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Successors of singular cardinals
Theorem (LH-Rinot)
Suppose that λ is a singular limit of supercompact cardinals.
Then χ(λ+) = cf(λ).
Moreover, if θ ∈ (cf(λ), λ) is a regular cardinal, then there is a
cofinality-preserving forcing extension in which χ(λ+) = θ.
Theorem (LH-Rinot)
Suppose that λ is a supercompact cardinal. In a Prikry extension
using a normal measure over λ, we have χ(λ+) = ℵ0.
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Successors of singular cardinals
Theorem (LH-Rinot)
Suppose that λ is a singular limit of supercompact cardinals.
Then χ(λ+) = cf(λ).
Moreover, if θ ∈ (cf(λ), λ) is a regular cardinal, then there is a
cofinality-preserving forcing extension in which χ(λ+) = θ.
Theorem (LH-Rinot)
Suppose that λ is a supercompact cardinal. In a Prikry extension
using a normal measure over λ, we have χ(λ+) = ℵ0.
Moreover, collapses can be interleaved into the Prikry forcing so
that, in the generic extension, λ = ℵω and χ(ℵω+1) = ℵ0.
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II. Unbounded colorings
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U(
(
(· · ·)
)
)
Definition
Suppose that µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the
existence of a coloring c : [κ]2 → θ such that,
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U(
(
(· · ·)
)
)
Definition
Suppose that µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the
existence of a coloring c : [κ]2 → θ such that, for every χ < χ,
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U(
(
(· · ·)
)
)
Definition
Suppose that µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the
existence of a coloring c : [κ]2 → θ such that, for every χ < χ,
for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets,
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U(
(
(· · ·)
)
)
Definition
Suppose that µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the
existence of a coloring c : [κ]2 → θ such that, for every χ < χ,
for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets,
and for every i < θ,
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U(
(
(· · ·)
)
)
Definition
Suppose that µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the
existence of a coloring c : [κ]2 → θ such that, for every χ < χ,
for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets,
and for every i < θ, there is B ⊆ A of size µ such that,
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U(
(
(· · ·)
)
)
Definition
Suppose that µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the
existence of a coloring c : [κ]2 → θ such that, for every χ < χ,
for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets,
and for every i < θ, there is B ⊆ A of size µ such that, for all
distinct a, b ∈ B, we have min(c[a × b]) ≥ i.
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U(
(
(· · ·)
)
)
Definition
Suppose that µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the
existence of a coloring c : [κ]2 → θ such that, for every χ < χ,
for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets,
and for every i < θ, there is B ⊆ A of size µ such that, for all
distinct a, b ∈ B, we have min(c[a × b]) ≥ i.
• U(κ, 2, θ, 2) is a strong negation of κ → (κ)2
θ
.
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U(
(
(· · ·)
)
)
Definition
Suppose that µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the
existence of a coloring c : [κ]2 → θ such that, for every χ < χ,
for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets,
and for every i < θ, there is B ⊆ A of size µ such that, for all
distinct a, b ∈ B, we have min(c[a × b]) ≥ i.
• U(κ, 2, θ, 2) is a strong negation of κ → (κ)2
θ
.
• If µ∗ ≥ µ and χ∗ ≥ χ, then U(κ, µ∗, θ, χ∗) implies
U(κ, µ, θ, χ).
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U(
(
(· · ·)
)
)
Definition
Suppose that µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the
existence of a coloring c : [κ]2 → θ such that, for every χ < χ,
for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets,
and for every i < θ, there is B ⊆ A of size µ such that, for all
distinct a, b ∈ B, we have min(c[a × b]) ≥ i.
• U(κ, 2, θ, 2) is a strong negation of κ → (κ)2
θ
.
• If µ∗ ≥ µ and χ∗ ≥ χ, then U(κ, µ∗, θ, χ∗) implies
U(κ, µ, θ, χ). There is no such monotonicity in the third
coordinate.
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U(
(
(· · ·)
)
)
Definition
Suppose that µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the
existence of a coloring c : [κ]2 → θ such that, for every χ < χ,
for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets,
and for every i < θ, there is B ⊆ A of size µ such that, for all
distinct a, b ∈ B, we have min(c[a × b]) ≥ i.
• U(κ, 2, θ, 2) is a strong negation of κ → (κ)2
θ
.
• If µ∗ ≥ µ and χ∗ ≥ χ, then U(κ, µ∗, θ, χ∗) implies
U(κ, µ, θ, χ). There is no such monotonicity in the third
coordinate.
• Instances of this principle are implicit in previous work of, for
instance, Galvin, Todorcevic, and Shelah.
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A first construction
Theorem (LH-Rinot)
Suppose that θ, χ < κ are regular cardinals and there are a
sequence Hi | i < θ of pairwise disjoint stationary subsets of
κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all
α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅.
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A first construction
Theorem (LH-Rinot)
Suppose that θ, χ < κ are regular cardinals and there are a
sequence Hi | i < θ of pairwise disjoint stationary subsets of
κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all
α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then
U(κ, κ, θ, χ) holds.
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A first construction
Theorem (LH-Rinot)
Suppose that θ, χ < κ are regular cardinals and there are a
sequence Hi | i < θ of pairwise disjoint stationary subsets of
κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all
α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then
U(κ, κ, θ, χ) holds.
Corollary
Any of the following statements entails U(κ, κ, θ, χ).
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A first construction
Theorem (LH-Rinot)
Suppose that θ, χ < κ are regular cardinals and there are a
sequence Hi | i < θ of pairwise disjoint stationary subsets of
κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all
α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then
U(κ, κ, θ, χ) holds.
Corollary
Any of the following statements entails U(κ, κ, θ, χ).
1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ).
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A first construction
Theorem (LH-Rinot)
Suppose that θ, χ < κ are regular cardinals and there are a
sequence Hi | i < θ of pairwise disjoint stationary subsets of
κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all
α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then
U(κ, κ, θ, χ) holds.
Corollary
Any of the following statements entails U(κ, κ, θ, χ).
1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ).
2 κ = λ+ and λ is regular.
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A first construction
Theorem (LH-Rinot)
Suppose that θ, χ < κ are regular cardinals and there are a
sequence Hi | i < θ of pairwise disjoint stationary subsets of
κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all
α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then
U(κ, κ, θ, χ) holds.
Corollary
Any of the following statements entails U(κ, κ, θ, χ).
1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ).
2 κ = λ+ and λ is regular.
3 κ = λ+ and cf(λ) = θ.
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A first construction
Theorem (LH-Rinot)
Suppose that θ, χ < κ are regular cardinals and there are a
sequence Hi | i < θ of pairwise disjoint stationary subsets of
κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all
α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then
U(κ, κ, θ, χ) holds.
Corollary
Any of the following statements entails U(κ, κ, θ, χ).
1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ).
2 κ = λ+ and λ is regular.
3 κ = λ+ and cf(λ) = θ.
4 (κ) holds.
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Further results
Theorem (LH-Rinot)
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and S ⊆ κ is a stationary set such that either
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Further results
Theorem (LH-Rinot)
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and S ⊆ κ is a stationary set such that either
1 S ⊆ cof(χ) and ♣(S) holds; or
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Further results
Theorem (LH-Rinot)
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and S ⊆ κ is a stationary set such that either
1 S ⊆ cof(χ) and ♣(S) holds; or
2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal.
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Further results
Theorem (LH-Rinot)
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and S ⊆ κ is a stationary set such that either
1 S ⊆ cof(χ) and ♣(S) holds; or
2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal.
Then U(κ, κ, θ, χ) holds.
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Further results
Theorem (LH-Rinot)
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and S ⊆ κ is a stationary set such that either
1 S ⊆ cof(χ) and ♣(S) holds; or
2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal.
Then U(κ, κ, θ, χ) holds.
Theorem (LH-Rinot)
Suppose that λ is singular, θ < λ is regular, and either
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Further results
Theorem (LH-Rinot)
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and S ⊆ κ is a stationary set such that either
1 S ⊆ cof(χ) and ♣(S) holds; or
2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal.
Then U(κ, κ, θ, χ) holds.
Theorem (LH-Rinot)
Suppose that λ is singular, θ < λ is regular, and either
1 2λ = λ+; or
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Further results
Theorem (LH-Rinot)
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and S ⊆ κ is a stationary set such that either
1 S ⊆ cof(χ) and ♣(S) holds; or
2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal.
Then U(κ, κ, θ, χ) holds.
Theorem (LH-Rinot)
Suppose that λ is singular, θ < λ is regular, and either
1 2λ = λ+; or
2 there is a collection of fewer than cf(λ)-many stationary
subsets of λ+ that do not reflect simultaneously.
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Further results
Theorem (LH-Rinot)
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and S ⊆ κ is a stationary set such that either
1 S ⊆ cof(χ) and ♣(S) holds; or
2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal.
Then U(κ, κ, θ, χ) holds.
Theorem (LH-Rinot)
Suppose that λ is singular, θ < λ is regular, and either
1 2λ = λ+; or
2 there is a collection of fewer than cf(λ)-many stationary
subsets of λ+ that do not reflect simultaneously.
Then U(λ+, λ+, θ, cf(λ)) holds.
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Consistency results
Theorem (LH-Rinot)
Suppose that κ is a weakly compact cardinal and θ < κ is regular.
Then there is a cofinality-preserving forcing extension in which
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Consistency results
Theorem (LH-Rinot)
Suppose that κ is a weakly compact cardinal and θ < κ is regular.
Then there is a cofinality-preserving forcing extension in which
1 U(κ, κ, θ, κ) holds;
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Consistency results
Theorem (LH-Rinot)
Suppose that κ is a weakly compact cardinal and θ < κ is regular.
Then there is a cofinality-preserving forcing extension in which
1 U(κ, κ, θ, κ) holds;
2 U(κ, 2, θ , θ+) fails for all regular θ ∈ κ \ {θ}.
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Consistency results
Theorem (LH-Rinot)
Suppose that κ is a weakly compact cardinal and θ < κ is regular.
Then there is a cofinality-preserving forcing extension in which
1 U(κ, κ, θ, κ) holds;
2 U(κ, 2, θ , θ+) fails for all regular θ ∈ κ \ {θ}.
Theorem (LH-Rinot)
Suppose that λ is a singular limit of supercompact cardinals.
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Consistency results
Theorem (LH-Rinot)
Suppose that κ is a weakly compact cardinal and θ < κ is regular.
Then there is a cofinality-preserving forcing extension in which
1 U(κ, κ, θ, κ) holds;
2 U(κ, 2, θ , θ+) fails for all regular θ ∈ κ \ {θ}.
Theorem (LH-Rinot)
Suppose that λ is a singular limit of supercompact cardinals.
1 U(λ+, 2, θ, cf(λ)+) fails for every regular θ ∈ λ \ {cf(λ)}.
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Consistency results
Theorem (LH-Rinot)
Suppose that κ is a weakly compact cardinal and θ < κ is regular.
Then there is a cofinality-preserving forcing extension in which
1 U(κ, κ, θ, κ) holds;
2 U(κ, 2, θ , θ+) fails for all regular θ ∈ κ \ {θ}.
Theorem (LH-Rinot)
Suppose that λ is a singular limit of supercompact cardinals.
1 U(λ+, 2, θ, cf(λ)+) fails for every regular θ ∈ λ \ {cf(λ)}.
2 Suppose that θ ∈ (cf(λ), λ) is regular. There is a
cofinality-preserving forcing extension in which
U(λ+, λ+, θ, λ+) holds but U(λ+, 2, θ , θ+) fails for all
regular θ ∈ λ \ {θ, cf(λ)}.
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Prikry forcing
Theorem (LH-Rinot)
Suppose that λ is a supercompact cardinal. In the Prikry
extension obtained using a normal measure over λ,
U(λ+, 2, θ, ℵ1) fails for all regular, uncountable θ < λ.
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Prikry forcing
Theorem (LH-Rinot)
Suppose that λ is a supercompact cardinal. In the Prikry
extension obtained using a normal measure over λ,
U(λ+, 2, θ, ℵ1) fails for all regular, uncountable θ < λ.
Moreover, one may interleave collapses in the Prikry forcing to
obtain a model in which λ = ℵω and U(ℵω+1, 2, ℵn, ℵ1) fails for
all 1 ≤ n < ω.
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III. Infinite productivity
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Productivity of Knasterness
Recall that, if κ is a regular cardinal, then the κ-Knaster property
is always finitely productive.
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Productivity of Knasterness
Recall that, if κ is a regular cardinal, then the κ-Knaster property
is always finitely productive.
Moreover, if κ is weakly compact, then the κ-Knaster property is
θ-productive for all θ < κ.
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Productivity of Knasterness
Recall that, if κ is a regular cardinal, then the κ-Knaster property
is always finitely productive.
Moreover, if κ is weakly compact, then the κ-Knaster property is
θ-productive for all θ < κ.
This leads naturally to the following question: For what values of
κ can the κ-Knaster property be even ℵ0-productive?
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Productivity of Knasterness
Recall that, if κ is a regular cardinal, then the κ-Knaster property
is always finitely productive.
Moreover, if κ is weakly compact, then the κ-Knaster property is
θ-productive for all θ < κ.
This leads naturally to the following question: For what values of
κ can the κ-Knaster property be even ℵ0-productive?
In particular, is it consistent that there is a successor cardinal κ
for which the κ-Knaster property is ℵ0-productive?
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Some previous results
Theorem (Cox-L¨
ucke)
If the existence of a weakly compact cardinal is consistent, then it
is consistent that there is an inaccessible, non-weakly compact
cardinal κ such that the κ-Knaster property is θ-productive for all
θ < κ.
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Some previous results
Theorem (Cox-L¨
ucke)
If the existence of a weakly compact cardinal is consistent, then it
is consistent that there is an inaccessible, non-weakly compact
cardinal κ such that the κ-Knaster property is θ-productive for all
θ < κ.
Theorem (LH-L¨
ucke)
Suppose that κ is a regular cardinal and the κ-Knaster property is
ℵ0-productive. Then κ is weakly compact in L.
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IV. Connections
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The C-sequence number and
unbounded colorings
Theorem (LH-Rinot)
Suppose that ℵ0 ≤ χ ≤ θ = cf(θ) < κ and there is a closed
witness to U(κ, 2, θ, χ).
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The C-sequence number and
unbounded colorings
Theorem (LH-Rinot)
Suppose that ℵ0 ≤ χ ≤ θ = cf(θ) < κ and there is a closed
witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ.
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The C-sequence number and
unbounded colorings
Theorem (LH-Rinot)
Suppose that ℵ0 ≤ χ ≤ θ = cf(θ) < κ and there is a closed
witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ.
Theorem (Todorcevic, LH-Rinot)
There is a closed witness to U(κ, κ, ℵ0, χ(κ)).
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The C-sequence number and
unbounded colorings
Theorem (LH-Rinot)
Suppose that ℵ0 ≤ χ ≤ θ = cf(θ) < κ and there is a closed
witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ.
Theorem (Todorcevic, LH-Rinot)
There is a closed witness to U(κ, κ, ℵ0, χ(κ)).
Theorem (LH-Rinot)
There is a closed witness to U(κ, κ, χ(κ), χ(κ)).
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The C-sequence number and
unbounded colorings
Theorem (LH-Rinot)
Suppose that ℵ0 ≤ χ ≤ θ = cf(θ) < κ and there is a closed
witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ.
Theorem (Todorcevic, LH-Rinot)
There is a closed witness to U(κ, κ, ℵ0, χ(κ)).
Theorem (LH-Rinot)
There is a closed witness to U(κ, κ, χ(κ), χ(κ)).
Conjecture (LH-Rinot)
For every regular θ < κ, U(κ, κ, θ, χ(κ)) holds.
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Unbounded colorings and
productivity
Lemma (LH-Rinot)
Suppose that θ ≤ χ < κ are infinite, regular cardinals, κ is
(< χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset
P such that
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Unbounded colorings and
productivity
Lemma (LH-Rinot)
Suppose that θ ≤ χ < κ are infinite, regular cardinals, κ is
(< χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset
P such that
• P is well-met and χ-directed closed with greatest lower
bounds;
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Unbounded colorings and
productivity
Lemma (LH-Rinot)
Suppose that θ ≤ χ < κ are infinite, regular cardinals, κ is
(< χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset
P such that
• P is well-met and χ-directed closed with greatest lower
bounds;
• Pτ is κ-Knaster for all τ < θ;
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Unbounded colorings and
productivity
Lemma (LH-Rinot)
Suppose that θ ≤ χ < κ are infinite, regular cardinals, κ is
(< χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset
P such that
• P is well-met and χ-directed closed with greatest lower
bounds;
• Pτ is κ-Knaster for all τ < θ;
• Pθ is not κ-c.c.
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Unbounded colorings and
productivity
Lemma (LH-Rinot)
Suppose that θ ≤ χ < κ are infinite, regular cardinals, κ is
(< χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset
P such that
• P is well-met and χ-directed closed with greatest lower
bounds;
• Pτ is κ-Knaster for all τ < θ;
• Pθ is not κ-c.c.
Corollary
For every infinite λ, the λ+-Knaster property is not ℵ0-productive.
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Productivity, cont.
Sketch of proof of corollary.
Recall that cf(λ) ≤ χ(λ+) and that U(λ+, λ+, ℵ0, χ(λ+)) holds.
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Productivity, cont.
Sketch of proof of corollary.
Recall that cf(λ) ≤ χ(λ+) and that U(λ+, λ+, ℵ0, χ(λ+)) holds.
Now apply the Lemma with θ = χ = ℵ0 (every infinite cardinal is
(< ℵ0)-inaccessible.)
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Productivity, cont.
Sketch of proof of corollary.
Recall that cf(λ) ≤ χ(λ+) and that U(λ+, λ+, ℵ0, χ(λ+)) holds.
Now apply the Lemma with θ = χ = ℵ0 (every infinite cardinal is
(< ℵ0)-inaccessible.)
Conjecture
Suppose κ is a regular, uncountable cardinal. Then the κ-Knaster
property is ℵ0-productive if and only if χ(κ) < ℵ0, i.e., if and only
if every C-sequence over κ is trivial.
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References
The work in this talk comes from the following two papers, both
jointly written with Assaf Rinot:
• Knaster and friends I: Closed colorings and precalibers.
Preprint available at
math.biu.ac.il/∼lambiec/knaster1.pdf
• Knaster and friends II: Subadditive colorings and the
C-sequence number. Coming soon!
All feedback is welcome!
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References
The work in this talk comes from the following two papers, both
jointly written with Assaf Rinot:
• Knaster and friends I: Closed colorings and precalibers.
Preprint available at
math.biu.ac.il/∼lambiec/knaster1.pdf
• Knaster and friends II: Subadditive colorings and the
C-sequence number. Coming soon!
All feedback is welcome!
Artwork: “Bands of Color in Four Directions” by Sol Lewitt
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Thank you!