# Unbounded functions and infinite productivity of the Knaster property

Set Theory, Model Theory and Applications. Eilat, 2018.

May 03, 2018

## Transcript

1. ### 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

3. ### 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.
4. ### 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.
5. ### 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.
6. ### 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.
7. ### 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.
8. ### 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.
9. ### 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.
10. ### 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,
11. ### 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,
12. ### 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,
13. ### 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,
14. ### 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}.
15. ### 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.

property if,
17. ### The κ-Knaster property Deﬁnition A poset P has the κ-Knaster

property if, for every A ⊆ P of size κ,
18. ### 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.
19. ### 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.
20. ### 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.
21. ### 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?
22. ### 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.)
23. ### 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?

25. ### 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.
26. ### 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.
27. ### 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.
28. ### 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.
29. ### 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?

31. ### U(κ, µ, θ, χ) Deﬁnition U(κ, µ, θ, χ) asserts

the existence of a coloring c : [κ]2 → θ such that,
32. ### 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,
33. ### 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,
34. ### 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.
35. ### 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.
36. ### 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
37. ### 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;
38. ### 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;
39. ### 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.
40. ### 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).
41. ### 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.
42. ### 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 .
43. ### 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.

45. ### 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.
46. ### 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.
47. ### 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.

49. ### 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.
50. ### 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.
51. ### 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}.
52. ### 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.
53. ### Walking along C-sequences Deﬁnition A C-sequence over κ is a

sequence Cα | α < κ such that
54. ### 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);
55. ### 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.
56. ### 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.
57. ### 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) = ∅.
58. ### 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)]).
59. ### 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 = ∅.
60. ### 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).
61. ### 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
62. ### 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)]).
63. ### 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).
64. ### 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).
65. ### 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.
66. ### 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.
67. ### 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.
68. ### 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.
69. ### 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 ).
70. ### 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).
71. ### Further results Theorem Suppose that λ is a singular cardinal,

θ < λ is a regular cardinal, and either
72. ### Further results Theorem Suppose that λ is a singular cardinal,

θ < λ is a regular cardinal, and either 1 2λ = λ+; or
73. ### 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.
74. ### 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)).
75. ### 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
76. ### 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
77. ### 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.
78. ### 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).
79. ### Negative results Proposition Suppose that κ is a weakly compact

cardinal. Then U(κ, 2, θ, 2) fails for every θ < κ.
80. ### 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.
81. ### 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)}.
82. ### 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?
83. ### 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)?
84. ### 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)?

86. ### The C-sequence number Theorem (Todorcevic) For every regular, uncountable cardinal

κ, the following are equivalent.
87. ### The C-sequence number Theorem (Todorcevic) For every regular, uncountable cardinal

κ, the following are equivalent. 1 κ is weakly compact.
88. ### 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.
89. ### 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.
90. ### 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.
91. ### Some basic facts Proposition The C-sequence number satisﬁes the following

properties. 1 χ(κ) ≤ κ.
92. ### 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.
93. ### 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.
94. ### 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).
95. ### 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.
96. ### 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.
97. ### 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.
98. ### 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.
99. ### 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).
100. ### 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.
101. ### 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.
102. ### 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.
103. ### 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
104. ### 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;
105. ### 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.
106. ### 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.
107. ### χ(κ) and closed colorings Lemma Suppose that ℵ0 ≤ χ

≤ θ = cf(θ) < κ,
108. ### 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).
109. ### 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.
110. ### 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)).
111. ### 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)).
112. ### 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)).

114. ### Subadditivity Deﬁnition We say that a coloring c : [κ]2

→ θ is subadditive if, for all α < β < γ < κ,
115. ### 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)};
116. ### 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)}.
117. ### 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.
118. ### 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).
119. ### 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)).
120. ### 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)).
121. ### 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)).
122. ### 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.
123. ### 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.
124. ### 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.
125. ### 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.
126. ### 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.
127. ### 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.
128. ### 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).
129. ### 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.
130. ### 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
131. ### 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;
132. ### 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<j Pi is κ-stationarily layered;
133. ### 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<j Pi is κ-stationarily layered; 3 i<θ Pi is not κ-c.c.
134. ### References The work in this talk comes from the following

two papers, both jointly written with Assaf Rinot:
135. ### 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
136. ### 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!
137. ### 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!
138. ### 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