work with Assaf Rinot) Chris Lambie-Hanson Department of Mathematics Bar-Ilan University Set Theory, Model Theory and Applications Eilat, Israel 25 April 2018
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.
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.
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.
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.
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.
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,
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,
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,
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,
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}.
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.
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.
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.
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?
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.)
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?
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.
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.
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.
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?
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,
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.
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.
< 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;
< 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;
< 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.
< 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).
< 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.
< 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 .
< 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.
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.
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.
· · ) that have certain closure properties. These closed functions are better-behaved than arbitrary witnesses to U(· · · ) and also arise naturally in practice.
· · ) 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.
· · ) 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}.
· · ) 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.
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.
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) = ∅.
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)]).
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 = ∅.
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).
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
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)]).
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).
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.
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.
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.
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.
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 ).
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).
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.
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)).
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
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
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.
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).
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.
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)}.
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?
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)?
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)?
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.
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.
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.
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).
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.
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.
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.
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.
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).
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.
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.
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
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;
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.
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.
≤ 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)).
≤ 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)).
→ 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)).
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)).
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.
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.
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.
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.
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.
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.
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.
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
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;
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<j Pi is κ-stationarily layered;
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<j Pi is κ-stationarily layered; 3 i<θ Pi is not κ-c.c.
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
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!
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!
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