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Unbounded functions and infinite
productivity of the Knaster property
(Joint work with Assaf Rinot)
Chris Lambie-Hanson
Department of Mathematics
Bar-Ilan University
Set Theory, Model Theory and Applications
Eilat, Israel
25 April 2018
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I. Productivity of chain conditions
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The κ-chain condition
Recall that a property is said to be productive if, whenever P and
Q satisfy the property, then P × Q satisfies the property as well.
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The κ-chain condition
Recall that a property is said to be productive if, whenever P and
Q satisfy the property, then P × Q satisfies the property as well.
The question as to whether the κ-chain condition (for posets) is
(or can consistently be) productive, where κ is a regular,
uncountable cardinal, has led to a great deal of set theoretic
research.
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The κ-chain condition
Recall that a property is said to be productive if, whenever P and
Q satisfy the property, then P × Q satisfies the property as well.
The question as to whether the κ-chain condition (for posets) is
(or can consistently be) productive, where κ is a regular,
uncountable cardinal, has led to a great deal of set theoretic
research.
• If MAℵ1
holds, then the ℵ1-chain condition is productive.
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The κ-chain condition
Recall that a property is said to be productive if, whenever P and
Q satisfy the property, then P × Q satisfies the property as well.
The question as to whether the κ-chain condition (for posets) is
(or can consistently be) productive, where κ is a regular,
uncountable cardinal, has led to a great deal of set theoretic
research.
• If MAℵ1
holds, then the ℵ1-chain condition is productive.
• (Shelah) If κ > ℵ1 is a successor cardinal, then the κ-chain
condition is not productive.
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The κ-chain condition
Recall that a property is said to be productive if, whenever P and
Q satisfy the property, then P × Q satisfies the property as well.
The question as to whether the κ-chain condition (for posets) is
(or can consistently be) productive, where κ is a regular,
uncountable cardinal, has led to a great deal of set theoretic
research.
• If MAℵ1
holds, then the ℵ1-chain condition is productive.
• (Shelah) If κ > ℵ1 is a successor cardinal, then the κ-chain
condition is not productive.
• (Rinot) If κ > ℵ1 is a regular cardinal and the κ-chain
condition is productive, then κ is weakly compact in L.
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The κ-chain condition
Recall that a property is said to be productive if, whenever P and
Q satisfy the property, then P × Q satisfies the property as well.
The question as to whether the κ-chain condition (for posets) is
(or can consistently be) productive, where κ is a regular,
uncountable cardinal, has led to a great deal of set theoretic
research.
• If MAℵ1
holds, then the ℵ1-chain condition is productive.
• (Shelah) If κ > ℵ1 is a successor cardinal, then the κ-chain
condition is not productive.
• (Rinot) If κ > ℵ1 is a regular cardinal and the κ-chain
condition is productive, then κ is weakly compact in L.
Conjecture (Todorcevic)
For every regular cardinal κ > ℵ1, the κ-chain condition is
productive iff κ is weakly compact.
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Pr1
(κ, κ, θ, χ)
Throughout the talk, κ will be a regular uncountable cardinal, and
θ, χ, and µ will be cardinals less than or equal to κ.
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Pr1
(κ, κ, θ, χ)
Throughout the talk, κ will be a regular uncountable cardinal, and
θ, χ, and µ will be cardinals less than or equal to κ.
Definition (Shelah)
Pr1(κ, κ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ
such that,
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Pr1
(κ, κ, θ, χ)
Throughout the talk, κ will be a regular uncountable cardinal, and
θ, χ, and µ will be cardinals less than or equal to κ.
Definition (Shelah)
Pr1(κ, κ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ
such that, for every χ < χ,
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Pr1
(κ, κ, θ, χ)
Throughout the talk, κ will be a regular uncountable cardinal, and
θ, χ, and µ will be cardinals less than or equal to κ.
Definition (Shelah)
Pr1(κ, κ, 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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Pr1
(κ, κ, θ, χ)
Throughout the talk, κ will be a regular uncountable cardinal, and
θ, χ, and µ will be cardinals less than or equal to κ.
Definition (Shelah)
Pr1(κ, κ, 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 color i < θ,
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Pr1
(κ, κ, θ, χ)
Throughout the talk, κ will be a regular uncountable cardinal, and
θ, χ, and µ will be cardinals less than or equal to κ.
Definition (Shelah)
Pr1(κ, κ, 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 color i < θ, there
are a, b ∈ A such that a < b and c[a × b] = {i}.
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Pr1
(κ, κ, θ, χ)
Throughout the talk, κ will be a regular uncountable cardinal, and
θ, χ, and µ will be cardinals less than or equal to κ.
Definition (Shelah)
Pr1(κ, κ, 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 color i < θ, there
are a, b ∈ A such that a < b and c[a × b] = {i}.
Lemma
If χ < κ are infinite, regular cardinals, κ is (< χ)-inaccessible,
and Pr1(κ, κ, 2, χ) holds, then there is a χ-directed closed, κ-c.c.
poset P such that P2 is not κ-c.c.
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The κ-Knaster property
Definition
A poset P has the κ-Knaster property if,
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The κ-Knaster property
Definition
A poset P has the κ-Knaster property if, for every A ⊆ P of size
κ,
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The κ-Knaster property
Definition
A poset P has the κ-Knaster property if, for every A ⊆ P of size
κ, there is B ⊆ A of size κ consisting of pairwise ≤P
-compatible
conditions.
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The κ-Knaster property
Definition
A poset P has the κ-Knaster property if, for every A ⊆ P of size
κ, there is B ⊆ A of size κ consisting of pairwise ≤P
-compatible
conditions.
The κ-Knaster property is clearly a strengthening of the κ-chain
condition.
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The κ-Knaster property
Definition
A poset P has the κ-Knaster property if, for every A ⊆ P of size
κ, there is B ⊆ A of size κ consisting of pairwise ≤P
-compatible
conditions.
The κ-Knaster property is clearly a strengthening of the κ-chain
condition. Moreover, the κ-Knaster property is always productive.
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The κ-Knaster property
Definition
A poset P has the κ-Knaster property if, for every A ⊆ P of size
κ, there is B ⊆ A of size κ consisting of pairwise ≤P
-compatible
conditions.
The κ-Knaster property is clearly a strengthening of the κ-chain
condition. Moreover, the κ-Knaster property is always productive.
But can we ask for more?
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The κ-Knaster property
Definition
A poset P has the κ-Knaster property if, for every A ⊆ P of size
κ, there is B ⊆ A of size κ consisting of pairwise ≤P
-compatible
conditions.
The κ-Knaster property is clearly a strengthening of the κ-chain
condition. Moreover, the κ-Knaster property is always productive.
But can we ask for more? Say that a property is θ-productive if,
whenever {Pi | i < θ} all have the property, then i<θ Pi has the
property as well. (All products here are full-support.)
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The κ-Knaster property
Definition
A poset P has the κ-Knaster property if, for every A ⊆ P of size
κ, there is B ⊆ A of size κ consisting of pairwise ≤P
-compatible
conditions.
The κ-Knaster property is clearly a strengthening of the κ-chain
condition. Moreover, the κ-Knaster property is always productive.
But can we ask for more? Say that a property is θ-productive if,
whenever {Pi | i < θ} all have the property, then i<θ Pi has the
property as well. (All products here are full-support.)
Question
Under what circumstances can the κ-Knaster property be
θ-productive, where θ < κ is an infinite, regular cardinal?
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Infinite productivity
Some known facts:
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Infinite productivity
Some known facts:
• If κ is weakly compact, then all κ-c.c. posets are in fact
κ-Knaster, and both properties are θ-productive for all θ < κ.
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Infinite productivity
Some known facts:
• If κ is weakly compact, then all κ-c.c. posets are in fact
κ-Knaster, and both properties are θ-productive for all θ < κ.
• (Cox-L¨
ucke) Assuming the consistency of a weakly compact
cardinal, it is consistent that there is an inaccessible cardinal
κ that is not weakly compact and such that the κ-Knaster
property is θ-productive for all θ < κ.
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Infinite productivity
Some known facts:
• If κ is weakly compact, then all κ-c.c. posets are in fact
κ-Knaster, and both properties are θ-productive for all θ < κ.
• (Cox-L¨
ucke) Assuming the consistency of a weakly compact
cardinal, it is consistent that there is an inaccessible cardinal
κ that is not weakly compact and such that the κ-Knaster
property is θ-productive for all θ < κ.
• (LH-L¨
ucke) If the κ-Knaster property is ℵ0-productive, then
κ is weakly compact in L.
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Infinite productivity
Some known facts:
• If κ is weakly compact, then all κ-c.c. posets are in fact
κ-Knaster, and both properties are θ-productive for all θ < κ.
• (Cox-L¨
ucke) Assuming the consistency of a weakly compact
cardinal, it is consistent that there is an inaccessible cardinal
κ that is not weakly compact and such that the κ-Knaster
property is θ-productive for all θ < κ.
• (LH-L¨
ucke) If the κ-Knaster property is ℵ0-productive, then
κ is weakly compact in L.
These leave open some specific questions, including the following,
which we will answer today.
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Infinite productivity
Some known facts:
• If κ is weakly compact, then all κ-c.c. posets are in fact
κ-Knaster, and both properties are θ-productive for all θ < κ.
• (Cox-L¨
ucke) Assuming the consistency of a weakly compact
cardinal, it is consistent that there is an inaccessible cardinal
κ that is not weakly compact and such that the κ-Knaster
property is θ-productive for all θ < κ.
• (LH-L¨
ucke) If the κ-Knaster property is ℵ0-productive, then
κ is weakly compact in L.
These leave open some specific questions, including the following,
which we will answer today.
Question
Is there consistently a successor cardinal κ such that the
κ-Knaster property is ℵ0-productive?
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U(κ, µ, θ, χ)
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U(κ, µ, θ, χ)
Definition
U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such
that,
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U(κ, µ, θ, χ)
Definition
U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such
that, for every χ < χ and A ⊆ [κ]χ consisting of κ-many
pairwise disjoint sets,
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U(κ, µ, θ, χ)
Definition
U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such
that, for every χ < χ and A ⊆ [κ]χ consisting of κ-many
pairwise disjoint sets, and for every color i < θ,
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U(κ, µ, θ, χ)
Definition
U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such
that, for every χ < χ and A ⊆ [κ]χ consisting of κ-many
pairwise disjoint sets, and for every color i < θ, there is B ⊆ A
of size µ such that, for all a < b, both from B, we have
min(c[a × b]) > i.
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U(κ, µ, θ, χ)
Definition
U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such
that, for every χ < χ and A ⊆ [κ]χ consisting of κ-many
pairwise disjoint sets, and for every color i < θ, there is B ⊆ A
of size µ such that, for all a < b, both from B, we have
min(c[a × b]) > i.
Note that, for all µ ≤ µ and χ ≤ χ, U(κ, µ, θ, χ) implies
U(κ, µ , θ, χ ), but there is no such monotonicity in the third
coordinate.
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Failure of infinite productivity
Lemma
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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Failure of infinite productivity
Lemma
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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Failure of infinite productivity
Lemma
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 (in fact, has precaliber κ) for all τ < θ;
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Failure of infinite productivity
Lemma
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 (in fact, has precaliber κ) for all τ < θ;
• Pθ is not κ-c.c.
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Failure of infinite productivity
Lemma
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 (in fact, has precaliber κ) for all τ < θ;
• Pθ is not κ-c.c.
Sketch of proof.
Let c : [κ]2 → θ witness U(κ, κ, θ, χ).
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Failure of infinite productivity
Lemma
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 (in fact, has precaliber κ) for all τ < θ;
• Pθ is not κ-c.c.
Sketch of proof.
Let c : [κ]2 → θ witness U(κ, κ, θ, χ). For all i < θ, let Pi be the
poset whose conditions are all sets x ∈ [κ]<χ such that
min(c“[x]2) > i, ordered by reverse inclusion.
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Failure of infinite productivity
Lemma
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 (in fact, has precaliber κ) for all τ < θ;
• Pθ is not κ-c.c.
Sketch of proof.
Let c : [κ]2 → θ witness U(κ, κ, θ, χ). For all i < θ, let Pi be the
poset whose conditions are all sets x ∈ [κ]<χ such that
min(c“[x]2) > i, ordered by reverse inclusion. Let P be the lottery
sum i<θ Pi .
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Failure of infinite productivity
Lemma
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 (in fact, has precaliber κ) for all τ < θ;
• Pθ is not κ-c.c.
Sketch of proof.
Let c : [κ]2 → θ witness U(κ, κ, θ, χ). For all i < θ, let Pi be the
poset whose conditions are all sets x ∈ [κ]<χ such that
min(c“[x]2) > i, ordered by reverse inclusion. Let P be the lottery
sum i<θ Pi . Now check that P works.
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Productivity at successor cardinals
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Productivity at successor cardinals
The principle U(· · · ) has been implicit in a variety of previous
work. Particularly notable for us is the following result of
Todorcevic.
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Productivity at successor cardinals
The principle U(· · · ) has been implicit in a variety of previous
work. Particularly notable for us is the following result of
Todorcevic.
Theorem (Todorcevic)
For every infinite cardinal λ, U(λ+, λ+, ℵ0, cf(λ)) holds.
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Productivity at successor cardinals
The principle U(· · · ) has been implicit in a variety of previous
work. Particularly notable for us is the following result of
Todorcevic.
Theorem (Todorcevic)
For every infinite cardinal λ, U(λ+, λ+, ℵ0, cf(λ)) holds.
Corollary
For every successor cardinal κ, the κ-Knaster property fails to be
ℵ0-productive.
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II. Closed colorings
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Closed colorings
We are particularly interested in witnesses to U(· · · ) that have
certain closure properties. These closed functions are
better-behaved than arbitrary witnesses to U(· · · ) and also arise
naturally in practice.
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Closed colorings
We are particularly interested in witnesses to U(· · · ) that have
certain closure properties. These closed functions are
better-behaved than arbitrary witnesses to U(· · · ) and also arise
naturally in practice.
Definition
Suppose that c : [κ]2 → θ is a coloring.
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Closed colorings
We are particularly interested in witnesses to U(· · · ) that have
certain closure properties. These closed functions are
better-behaved than arbitrary witnesses to U(· · · ) and also arise
naturally in practice.
Definition
Suppose that c : [κ]2 → θ is a coloring.
1 For all β < κ and all i < θ, let
Dc
≤i
(β) := {α < β | c(α, β) ≤ i}.
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Closed colorings
We are particularly interested in witnesses to U(· · · ) that have
certain closure properties. These closed functions are
better-behaved than arbitrary witnesses to U(· · · ) and also arise
naturally in practice.
Definition
Suppose that c : [κ]2 → θ is a coloring.
1 For all β < κ and all i < θ, let
Dc
≤i
(β) := {α < β | c(α, β) ≤ i}.
2 We say that c is closed if Dc
≤i
(β) is closed as a subset of β
for all β < κ and all i < θ.
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Walking along C-sequences
Definition
A C-sequence over κ is a sequence Cα | α < κ such that
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Walking along C-sequences
Definition
A C-sequence over κ is a sequence Cα | α < κ such that
• Cα is a club in α for all α ∈ acc(κ);
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Walking along C-sequences
Definition
A C-sequence over κ is a sequence Cα | α < κ such that
• Cα is a club in α for all α ∈ acc(κ);
• Cα+1 = {α} for all α < κ.
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Walking along C-sequences
Definition
A C-sequence over κ is a sequence Cα | α < κ such that
• Cα is a club in α for all α ∈ acc(κ);
• Cα+1 = {α} for all α < κ.
Given a C-sequence C over κ, we will perform walks along C.
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Walking along C-sequences
Definition
A C-sequence over κ is a sequence Cα | α < κ such that
• Cα is a club in α for all α ∈ acc(κ);
• Cα+1 = {α} for all α < κ.
Given a C-sequence C over κ, we will perform walks along C. We
define the upper trace function tr : [κ]2 → [κ]<ω recursively by
tr(α, β) = {β} ∪ tr(α, min(Cβ \ α)),
subject to the boundary condition tr(α, α) = ∅.
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Walking along C-sequences
Definition
A C-sequence over κ is a sequence Cα | α < κ such that
• Cα is a club in α for all α ∈ acc(κ);
• Cα+1 = {α} for all α < κ.
Given a C-sequence C over κ, we will perform walks along C. We
define the upper trace function tr : [κ]2 → [κ]<ω recursively by
tr(α, β) = {β} ∪ tr(α, min(Cβ \ α)),
subject to the boundary condition tr(α, α) = ∅.
Our primary strategy for defining closed witnesses to U(κ, µ, θ, χ)
is to isolate an appropriate function h : κ → θ and define a
coloring c : [κ]2 → θ by letting
c(α, β) = max(h[tr(α, β)]).
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A first construction
Theorem
Suppose that θ < κ are regular cardinals and there are a
sequence Hi | i < θ of pairwise disjoint stationary subsets of κ
and a C-sequence C = Cα | α < κ such that, for all α < κ and
all sufficiently large i < θ, acc(Cα) ∩ Hi = ∅.
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A first construction
Theorem
Suppose that θ < κ are regular cardinals and there are a
sequence Hi | i < θ of pairwise disjoint stationary subsets of κ
and a C-sequence C = Cα | α < κ such that, for all α < κ and
all sufficiently large i < θ, acc(Cα) ∩ Hi = ∅. Then there is a
closed coloring c : [κ]2 → θ such that, for all regular χ < κ, if
Hi ∩ cof(≥ χ) is stationary for unboundedly many i < θ, then c
witnesses U(κ, κ, θ, χ).
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A first construction
Theorem
Suppose that θ < κ are regular cardinals and there are a
sequence Hi | i < θ of pairwise disjoint stationary subsets of κ
and a C-sequence C = Cα | α < κ such that, for all α < κ and
all sufficiently large i < θ, acc(Cα) ∩ Hi = ∅. Then there is a
closed coloring c : [κ]2 → θ such that, for all regular χ < κ, if
Hi ∩ cof(≥ χ) is stationary for unboundedly many i < θ, then c
witnesses U(κ, κ, θ, χ).
Sketch of proof.
Walk along C, and define h : κ → θ and c : [κ]2 → θ by letting
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A first construction
Theorem
Suppose that θ < κ are regular cardinals and there are a
sequence Hi | i < θ of pairwise disjoint stationary subsets of κ
and a C-sequence C = Cα | α < κ such that, for all α < κ and
all sufficiently large i < θ, acc(Cα) ∩ Hi = ∅. Then there is a
closed coloring c : [κ]2 → θ such that, for all regular χ < κ, if
Hi ∩ cof(≥ χ) is stationary for unboundedly many i < θ, then c
witnesses U(κ, κ, θ, χ).
Sketch of proof.
Walk along C, and define h : κ → θ and c : [κ]2 → θ by letting
h(η) = sup{i < θ | η ∈ Hi or acc(Cη) ∩ Hi = ∅};
c(α, β) = max(h[tr(α, β)]).
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Corollaries
Corollary
Suppose that θ, χ < κ are regular cardinals. Any one of the
following entails the existence of a closed witness to U(κ, κ, θ, χ).
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Corollaries
Corollary
Suppose that θ, χ < κ are regular cardinals. Any one of the
following entails the existence of a closed witness to U(κ, κ, θ, χ).
1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ).
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Corollaries
Corollary
Suppose that θ, χ < κ are regular cardinals. Any one of the
following entails the existence of a closed witness to 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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Corollaries
Corollary
Suppose that θ, χ < κ are regular cardinals. Any one of the
following entails the existence of a closed witness to 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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Corollaries
Corollary
Suppose that θ, χ < κ are regular cardinals. Any one of the
following entails the existence of a closed witness to 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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Corollaries
Corollary
Suppose that θ, χ < κ are regular cardinals. Any one of the
following entails the existence of a closed witness to 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.
Sketch of Proof of (3).
Let λi | i < θ be an increasing sequence of regular cardinals
converging to λ.
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Corollaries
Corollary
Suppose that θ, χ < κ are regular cardinals. Any one of the
following entails the existence of a closed witness to 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.
Sketch of Proof of (3).
Let λi | i < θ be an increasing sequence of regular cardinals
converging to λ. Let Hi = κ ∩ cof(λi ).
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Corollaries
Corollary
Suppose that θ, χ < κ are regular cardinals. Any one of the
following entails the existence of a closed witness to 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.
Sketch of Proof of (3).
Let λi | i < θ be an increasing sequence of regular cardinals
converging to λ. Let Hi = κ ∩ cof(λi ). Let Cα | α < κ be a
C-sequence such that otp(Cα) = cf(α) for all α ∈ acc(κ).
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Further results
Theorem
Suppose that λ is a singular cardinal, θ < λ is a regular cardinal,
and either
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Further results
Theorem
Suppose that λ is a singular cardinal, θ < λ is a regular cardinal,
and either
1 2λ = λ+; or
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Further results
Theorem
Suppose that λ is a singular cardinal, θ < λ is a regular cardinal,
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
Suppose that λ is a singular cardinal, θ < λ is a regular cardinal,
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 there is a closed witness to U(λ+, λ+, θ, cf(λ)).
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Further results
Theorem
Suppose that λ is a singular cardinal, θ < λ is a regular cardinal,
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 there is a closed witness to U(λ+, λ+, θ, cf(λ)).
Theorem
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and there is a stationary S ⊆ κ such that either
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Further results
Theorem
Suppose that λ is a singular cardinal, θ < λ is a regular cardinal,
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 there is a closed witness to U(λ+, λ+, θ, cf(λ)).
Theorem
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and there is a stationary S ⊆ κ such that either
1 S ⊆ cof(χ) and ♣(S) holds; or
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Further results
Theorem
Suppose that λ is a singular cardinal, θ < λ is a regular cardinal,
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 there is a closed witness to U(λ+, λ+, θ, cf(λ)).
Theorem
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and there is a stationary S ⊆ κ 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
Suppose that λ is a singular cardinal, θ < λ is a regular cardinal,
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 there is a closed witness to U(λ+, λ+, θ, cf(λ)).
Theorem
Suppose that κ is an inaccessible cardinal, θ, χ < κ are regular
cardinals, and there is a stationary S ⊆ κ such that either
1 S ⊆ cof(χ) and ♣(S) holds; or
2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal.
Then there is a closed witness to U(κ, κ, θ, χ).
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Negative results
Proposition
Suppose that κ is a weakly compact cardinal. Then U(κ, 2, θ, 2)
fails for every θ < κ.
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Negative results
Proposition
Suppose that κ is a weakly compact cardinal. Then U(κ, 2, θ, 2)
fails for every θ < κ.
We conjecture that this in fact characterizes weak compactness
among regular, uncountable cardinals.
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Negative results
Proposition
Suppose that κ is a weakly compact cardinal. Then U(κ, 2, θ, 2)
fails for every θ < κ.
We conjecture that this in fact characterizes weak compactness
among regular, uncountable cardinals.
Theorem
Suppose that λ is a singular limit of strongly compact cardinals.
Then U(λ+, 2, θ, cf(λ)+) fails for every regular θ ∈ λ \ {cf(λ)}.
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Negative results
Proposition
Suppose that κ is a weakly compact cardinal. Then U(κ, 2, θ, 2)
fails for every θ < κ.
We conjecture that this in fact characterizes weak compactness
among regular, uncountable cardinals.
Theorem
Suppose that λ is a singular limit of strongly compact cardinals.
Then U(λ+, 2, θ, cf(λ)+) fails for every regular θ ∈ λ \ {cf(λ)}.
Question
Can there consistently be a nontrivial failure of U(· · · ) at ℵω+1?
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Negative results
Proposition
Suppose that κ is a weakly compact cardinal. Then U(κ, 2, θ, 2)
fails for every θ < κ.
We conjecture that this in fact characterizes weak compactness
among regular, uncountable cardinals.
Theorem
Suppose that λ is a singular limit of strongly compact cardinals.
Then U(λ+, 2, θ, cf(λ)+) fails for every regular θ ∈ λ \ {cf(λ)}.
Question
Can there consistently be a nontrivial failure of U(· · · ) at ℵω+1?
For example, U(ℵω+1, 2, ℵ1, ℵ1)?
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Negative results
Proposition
Suppose that κ is a weakly compact cardinal. Then U(κ, 2, θ, 2)
fails for every θ < κ.
We conjecture that this in fact characterizes weak compactness
among regular, uncountable cardinals.
Theorem
Suppose that λ is a singular limit of strongly compact cardinals.
Then U(λ+, 2, θ, cf(λ)+) fails for every regular θ ∈ λ \ {cf(λ)}.
Question
Can there consistently be a nontrivial failure of U(· · · ) at ℵω+1?
For example, U(ℵω+1, 2, ℵ1, ℵ1)? Or U(ℵω+1, ℵω+1, ℵ1, ℵ1)?
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III. The C-sequence number
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The C-sequence number
Theorem (Todorcevic)
For every regular, uncountable cardinal κ, the following are
equivalent.
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The C-sequence number
Theorem (Todorcevic)
For every regular, uncountable cardinal κ, the following are
equivalent.
1 κ is weakly compact.
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The C-sequence number
Theorem (Todorcevic)
For every regular, uncountable cardinal κ, the following are
equivalent.
1 κ is weakly compact.
2 For every C-sequence Cα | α < κ , there is an unbounded
D ⊆ κ such that, for every α < κ, there is β < κ for which
D ∩ α = Cβ ∩ α.
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The C-sequence number
Theorem (Todorcevic)
For every regular, uncountable cardinal κ, the following are
equivalent.
1 κ is weakly compact.
2 For every C-sequence Cα | α < κ , there is an unbounded
D ⊆ κ such that, for every α < κ, there is β < κ for which
D ∩ α = Cβ ∩ α.
Definition (The C-sequence number)
For every regular, uncountable cardinal κ, let χ(κ) = 0 if κ is
weakly compact.
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The C-sequence number
Theorem (Todorcevic)
For every regular, uncountable cardinal κ, the following are
equivalent.
1 κ is weakly compact.
2 For every C-sequence Cα | α < κ , there is an unbounded
D ⊆ κ such that, for every α < κ, there is β < κ for which
D ∩ α = Cβ ∩ α.
Definition (The C-sequence number)
For every regular, uncountable cardinal κ, let χ(κ) = 0 if κ is
weakly compact. Otherwise, let χ(κ) be the least cardinal χ such
that, for every C-sequence Cα | α < κ , there is an unbounded
D ⊆ κ such that, for every α < κ, there is b ∈ [κ]χ for which
D ∩ α ⊆ β∈b
Cβ.
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Some basic facts
Proposition
The C-sequence number satisfies the following properties.
1 χ(κ) ≤ κ.
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Some basic facts
Proposition
The C-sequence number satisfies the following properties.
1 χ(κ) ≤ κ.
2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ.
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Some basic facts
Proposition
The C-sequence number satisfies the following properties.
1 χ(κ) ≤ κ.
2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In
particular, if λ is regular, then χ(λ+) = λ.
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Some basic facts
Proposition
The C-sequence number satisfies the following properties.
1 χ(κ) ≤ κ.
2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In
particular, if λ is regular, then χ(λ+) = λ.
3 If λ is a singular limit of supercompact cardinals, then
χ(λ+) = cf(λ).
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Some basic facts
Proposition
The C-sequence number satisfies the following properties.
1 χ(κ) ≤ κ.
2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In
particular, if λ is regular, then χ(λ+) = λ.
3 If λ is a singular limit of supercompact cardinals, then
χ(λ+) = cf(λ).
4 If χ(κ) > 1, then χ(κ) ≥ ℵ0.
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Some basic facts
Proposition
The C-sequence number satisfies the following properties.
1 χ(κ) ≤ κ.
2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In
particular, if λ is regular, then χ(λ+) = λ.
3 If λ is a singular limit of supercompact cardinals, then
χ(λ+) = cf(λ).
4 If χ(κ) > 1, then χ(κ) ≥ ℵ0.
5 Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
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Some basic facts
Proposition
The C-sequence number satisfies the following properties.
1 χ(κ) ≤ κ.
2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In
particular, if λ is regular, then χ(λ+) = λ.
3 If λ is a singular limit of supercompact cardinals, then
χ(λ+) = cf(λ).
4 If χ(κ) > 1, then χ(κ) ≥ ℵ0.
5 Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
6 If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0.
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Some basic facts
Proposition
The C-sequence number satisfies the following properties.
1 χ(κ) ≤ κ.
2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In
particular, if λ is regular, then χ(λ+) = λ.
3 If λ is a singular limit of supercompact cardinals, then
χ(λ+) = cf(λ).
4 If χ(κ) > 1, then χ(κ) ≥ ℵ0.
5 Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
6 If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0.
7 If (κ, < ω) holds, then χ(κ) is as large as possible.
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Some basic facts
Proposition
The C-sequence number satisfies the following properties.
1 χ(κ) ≤ κ.
2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In
particular, if λ is regular, then χ(λ+) = λ.
3 If λ is a singular limit of supercompact cardinals, then
χ(λ+) = cf(λ).
4 If χ(κ) > 1, then χ(κ) ≥ ℵ0.
5 Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
6 If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0.
7 If (κ, < ω) holds, then χ(κ) is as large as possible. I.e.,
χ(κ) = sup(κ ∩ Reg).
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Some basic facts
Proposition
Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
Sketch of the Proof.
Suppose that S ⊆ κ ∩ cof(> χ(κ)) is stationary and S does not
reflect.
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Some basic facts
Proposition
Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
Sketch of the Proof.
Suppose that S ⊆ κ ∩ cof(> χ(κ)) is stationary and S does not
reflect. Let C = Cα | α < κ be a C-sequence such that
acc(Cα) ∩ S = ∅ for all α < κ.
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Some basic facts
Proposition
Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
Sketch of the Proof.
Suppose that S ⊆ κ ∩ cof(> χ(κ)) is stationary and S does not
reflect. Let C = Cα | α < κ be a C-sequence such that
acc(Cα) ∩ S = ∅ for all α < κ. Let D ⊆ κ witness the value of
χ(κ) for C, i.e., for all α < κ, there is b ∈ [κ]χ(κ) such that
D ∩ α ⊆ β∈b
Cβ.
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Some basic facts
Proposition
Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
Sketch of the Proof.
Suppose that S ⊆ κ ∩ cof(> χ(κ)) is stationary and S does not
reflect. Let C = Cα | α < κ be a C-sequence such that
acc(Cα) ∩ S = ∅ for all α < κ. Let D ⊆ κ witness the value of
χ(κ) for C, i.e., for all α < κ, there is b ∈ [κ]χ(κ) such that
D ∩ α ⊆ β∈b
Cβ. Then we can find a limit point δ of D such
that
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Some basic facts
Proposition
Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
Sketch of the Proof.
Suppose that S ⊆ κ ∩ cof(> χ(κ)) is stationary and S does not
reflect. Let C = Cα | α < κ be a C-sequence such that
acc(Cα) ∩ S = ∅ for all α < κ. Let D ⊆ κ witness the value of
χ(κ) for C, i.e., for all α < κ, there is b ∈ [κ]χ(κ) such that
D ∩ α ⊆ β∈b
Cβ. Then we can find a limit point δ of D such
that
• δ ∈ S;
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Some basic facts
Proposition
Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
Sketch of the Proof.
Suppose that S ⊆ κ ∩ cof(> χ(κ)) is stationary and S does not
reflect. Let C = Cα | α < κ be a C-sequence such that
acc(Cα) ∩ S = ∅ for all α < κ. Let D ⊆ κ witness the value of
χ(κ) for C, i.e., for all α < κ, there is b ∈ [κ]χ(κ) such that
D ∩ α ⊆ β∈b
Cβ. Then we can find a limit point δ of D such
that
• δ ∈ S;
• there is β ∈ κ \ (δ + 1) such that δ is a limit point of Cβ.
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Some basic facts
Proposition
Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
Sketch of the Proof.
Suppose that S ⊆ κ ∩ cof(> χ(κ)) is stationary and S does not
reflect. Let C = Cα | α < κ be a C-sequence such that
acc(Cα) ∩ S = ∅ for all α < κ. Let D ⊆ κ witness the value of
χ(κ) for C, i.e., for all α < κ, there is b ∈ [κ]χ(κ) such that
D ∩ α ⊆ β∈b
Cβ. Then we can find a limit point δ of D such
that
• δ ∈ S;
• there is β ∈ κ \ (δ + 1) such that δ is a limit point of Cβ.
But then δ ∈ acc(Cβ) ∩ S, which is a contradiction.
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χ(κ) and closed colorings
Lemma
Suppose that ℵ0 ≤ χ ≤ θ = cf(θ) < κ,
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χ(κ) and closed colorings
Lemma
Suppose that ℵ0 ≤ χ ≤ θ = cf(θ) < κ, and suppose that there is
a closed witness to U(κ, 2, θ, χ).
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χ(κ) and closed colorings
Lemma
Suppose that ℵ0 ≤ χ ≤ θ = cf(θ) < κ, and suppose that there is
a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ.
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χ(κ) and closed colorings
Lemma
Suppose that ℵ0 ≤ χ ≤ θ = cf(θ) < κ, and suppose that there is
a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ.
Theorem
There is a closed witness to U(κ, κ, ω, χ(κ)).
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χ(κ) and closed colorings
Lemma
Suppose that ℵ0 ≤ χ ≤ θ = cf(θ) < κ, and suppose that there is
a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ.
Theorem
There is a closed witness to U(κ, κ, ω, χ(κ)).
Theorem
If χ(κ) ≥ ℵ0, there is a closed witness to U(κ, κ, χ(κ), χ(κ)).
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χ(κ) and closed colorings
Lemma
Suppose that ℵ0 ≤ χ ≤ θ = cf(θ) < κ, and suppose that there is
a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ.
Theorem
There is a closed witness to U(κ, κ, ω, χ(κ)).
Theorem
If χ(κ) ≥ ℵ0, there is a closed witness to U(κ, κ, χ(κ), χ(κ)).
As far as we know, it could be a ZFC theorem that, for all infinite,
regular θ < κ, there is a (closed) witness to U(κ, κ, θ, χ(κ)).
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IV. Subadditive colorings
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Subadditivity
Definition
We say that a coloring c : [κ]2 → θ is subadditive if, for all
α < β < γ < κ,
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Subadditivity
Definition
We say that a coloring c : [κ]2 → θ is subadditive if, for all
α < β < γ < κ,
1 c(α, γ) ≤ max{c(α, β), c(β, γ)};
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Subadditivity
Definition
We say that a coloring c : [κ]2 → θ is subadditive if, for all
α < β < γ < κ,
1 c(α, γ) ≤ max{c(α, β), c(β, γ)};
2 c(α, β) ≤ max{c(α, γ), c(β, γ)}.
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Subadditivity
Definition
We say that a coloring c : [κ]2 → θ is subadditive if, for all
α < β < γ < κ,
1 c(α, γ) ≤ max{c(α, β), c(β, γ)};
2 c(α, β) ≤ max{c(α, γ), c(β, γ)}.
Closed, subadditive witnesses to U(· · · ) behave particularly nicely.
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Subadditivity
Definition
We say that a coloring c : [κ]2 → θ is subadditive if, for all
α < β < γ < κ,
1 c(α, γ) ≤ max{c(α, β), c(β, γ)};
2 c(α, β) ≤ max{c(α, γ), c(β, γ)}.
Closed, subadditive witnesses to U(· · · ) behave particularly nicely.
Lemma
Suppose that θ < κ is regular and c : [κ]2 → θ is a closed,
subadditive witness to U(κ, 2, θ, 2).
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Subadditivity
Definition
We say that a coloring c : [κ]2 → θ is subadditive if, for all
α < β < γ < κ,
1 c(α, γ) ≤ max{c(α, β), c(β, γ)};
2 c(α, β) ≤ max{c(α, γ), c(β, γ)}.
Closed, subadditive witnesses to U(· · · ) behave particularly nicely.
Lemma
Suppose that θ < κ is regular and c : [κ]2 → θ is a closed,
subadditive witness to U(κ, 2, θ, 2). Then c is in fact a witness to
U(κ, κ, θ, sup(κ ∩ Reg)).
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Subadditivity and squares
Theorem
If (κ) holds, then, for every regular θ < κ, there is a closed,
subadditive witness to U(κ, κ, θ, sup(κ ∩ Reg)).
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Subadditivity and squares
Theorem
If (κ) holds, then, for every regular θ < κ, there is a closed,
subadditive witness to U(κ, κ, θ, sup(κ ∩ Reg)).
In fact, by a result of LH and L¨
ucke, (κ) implies ind(κ, θ), and
our result states that ind(κ, θ) implies the existence of a closed,
subadditive witness to U(κ, κ, θ, sup(κ ∩ Reg)).
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Subadditivity and squares
Theorem
If (κ) holds, then, for every regular θ < κ, there is a closed,
subadditive witness to U(κ, κ, θ, sup(κ ∩ Reg)).
In fact, by a result of LH and L¨
ucke, (κ) implies ind(κ, θ), and
our result states that ind(κ, θ) implies the existence of a closed,
subadditive witness to U(κ, κ, θ, sup(κ ∩ Reg)).
By results obtained independently by Shani and LH, (κ, 2) (or
even λ,2, if κ = λ+) does not suffice to obtain the conclusion of
the Theorem.
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Sequential fans
Given infinite cardinals θ ≤ κ, the sequential fan Fκ,θ is a
topological space with underlying set (κ × θ) ∪ {∗}. Each point in
κ × θ is isolated, and a basic open set of ∗ is a set of the form
Uf := {(α, i) | α < κ, i ≥ f (α)} ∪ {∗}, where f : κ → θ.
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Sequential fans
Given infinite cardinals θ ≤ κ, the sequential fan Fκ,θ is a
topological space with underlying set (κ × θ) ∪ {∗}. Each point in
κ × θ is isolated, and a basic open set of ∗ is a set of the form
Uf := {(α, i) | α < κ, i ≥ f (α)} ∪ {∗}, where f : κ → θ.
There are interesting questions regarding the tightness of the
square of the sequential fan, particularly at ℵ2, that are
connected to questions about compactness for collectionwise
Hausdorffness in first countable spaces.
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Sequential fans
Given infinite cardinals θ ≤ κ, the sequential fan Fκ,θ is a
topological space with underlying set (κ × θ) ∪ {∗}. Each point in
κ × θ is isolated, and a basic open set of ∗ is a set of the form
Uf := {(α, i) | α < κ, i ≥ f (α)} ∪ {∗}, where f : κ → θ.
There are interesting questions regarding the tightness of the
square of the sequential fan, particularly at ℵ2, that are
connected to questions about compactness for collectionwise
Hausdorffness in first countable spaces. The following observation
is essentially due to Todorcevic.
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Sequential fans
Given infinite cardinals θ ≤ κ, the sequential fan Fκ,θ is a
topological space with underlying set (κ × θ) ∪ {∗}. Each point in
κ × θ is isolated, and a basic open set of ∗ is a set of the form
Uf := {(α, i) | α < κ, i ≥ f (α)} ∪ {∗}, where f : κ → θ.
There are interesting questions regarding the tightness of the
square of the sequential fan, particularly at ℵ2, that are
connected to questions about compactness for collectionwise
Hausdorffness in first countable spaces. The following observation
is essentially due to Todorcevic.
Fact
If there is a subadditive witness to U(κ, 2, θ, 2), then t(F2
κ,θ
) = κ.
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Sequential fans
Given infinite cardinals θ ≤ κ, the sequential fan Fκ,θ is a
topological space with underlying set (κ × θ) ∪ {∗}. Each point in
κ × θ is isolated, and a basic open set of ∗ is a set of the form
Uf := {(α, i) | α < κ, i ≥ f (α)} ∪ {∗}, where f : κ → θ.
There are interesting questions regarding the tightness of the
square of the sequential fan, particularly at ℵ2, that are
connected to questions about compactness for collectionwise
Hausdorffness in first countable spaces. The following observation
is essentially due to Todorcevic.
Fact
If there is a subadditive witness to U(κ, 2, θ, 2), then t(F2
κ,θ
) = κ.
The consistency of t(F2
ℵ2,ℵ0
) < ℵ2 is a major open question.
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Stationarily layered posets
Definition (Cox)
A poset P is κ-stationarily layered if the collection of regular
suborders of P of size less than κ is stationary in Pκ(P).
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Stationarily layered posets
Definition (Cox)
A poset P is κ-stationarily layered if the collection of regular
suborders of P of size less than κ is stationary in Pκ(P).
If P is κ-stationarily layered, then P has precaliber κ.
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Stationarily layered posets
Definition (Cox)
A poset P is κ-stationarily layered if the collection of regular
suborders of P of size less than κ is stationary in Pκ(P).
If P is κ-stationarily layered, then P has precaliber κ.
Theorem
Suppose that θ < κ are regular, κ is θ-inaccessible, and there is a
closed, subadditive witness to U(κ, 2, θ, 2). Then there are posets
{Pi | i < θ} such that
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Stationarily layered posets
Definition (Cox)
A poset P is κ-stationarily layered if the collection of regular
suborders of P of size less than κ is stationary in Pκ(P).
If P is κ-stationarily layered, then P has precaliber κ.
Theorem
Suppose that θ < κ are regular, κ is θ-inaccessible, and there is a
closed, subadditive witness to U(κ, 2, θ, 2). Then there are posets
{Pi | i < θ} such that
1 each Pi is well-met and θ+-directed closed with greatest
lower bounds;
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Stationarily layered posets
Definition (Cox)
A poset P is κ-stationarily layered if the collection of regular
suborders of P of size less than κ is stationary in Pκ(P).
If P is κ-stationarily layered, then P has precaliber κ.
Theorem
Suppose that θ < κ are regular, κ is θ-inaccessible, and there is a
closed, subadditive witness to U(κ, 2, θ, 2). Then there are posets
{Pi | i < θ} such that
1 each Pi is well-met and θ+-directed closed with greatest
lower bounds;
2 for all j < θ, i
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Stationarily layered posets
Definition (Cox)
A poset P is κ-stationarily layered if the collection of regular
suborders of P of size less than κ is stationary in Pκ(P).
If P is κ-stationarily layered, then P has precaliber κ.
Theorem
Suppose that θ < κ are regular, κ is θ-inaccessible, and there is a
closed, subadditive witness to U(κ, 2, θ, 2). Then there are posets
{Pi | i < θ} such that
1 each Pi is well-met and θ+-directed closed with greatest
lower bounds;
2 for all j < θ, i
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References
The work in this talk comes from the following two papers, both
jointly written with Assaf Rinot:
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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
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Slide 136 text
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!
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Slide 137 text
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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Slide 138 text
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!
All artwork by Clifford Styll
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Thank you!