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Unbounded colorings and the C-sequence number

Unbounded colorings and the C-sequence number

18cc56b609c6a4b0968e856e3793cba4?s=128

Chris Lambie-Hanson

July 03, 2018
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  1. Unbounded colorings and the C-sequence number (Joint work with Assaf

    Rinot) Chris Lambie-Hanson Department of Mathematics Bar-Ilan University SETTOP Novi Sad, Serbia 2 July 2018
  2. Table of contents ∅. Three properties of weakly compact cardinals

  3. Table of contents ∅. Three properties of weakly compact cardinals

    I. Introducing the C-sequence number
  4. Table of contents ∅. Three properties of weakly compact cardinals

    I. Introducing the C-sequence number II. Unbounded pair-colorings
  5. Table of contents ∅. Three properties of weakly compact cardinals

    I. Introducing the C-sequence number II. Unbounded pair-colorings III. Infinite productivity of strong chain conditions
  6. Table of contents ∅. Three properties of weakly compact cardinals

    I. Introducing the C-sequence number II. Unbounded pair-colorings III. Infinite productivity of strong chain conditions IV. Finding connections
  7. ∅ ∅ ∅. Weakly compact cardinals

  8. WCCs and C-sequences Definition (Todorcevic) Suppose that κ is a

    regular, uncountable cardinal. A C-sequence (over κ) is a sequence Cα | α < κ such that, for all limit α < κ, Cα is a club in α.
  9. WCCs and C-sequences Definition (Todorcevic) Suppose that κ is a

    regular, uncountable cardinal. A C-sequence (over κ) is a sequence Cα | α < κ such that, for all limit α < κ, Cα is a club in α. The C-sequence is trivial if there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α ⊆ Cβ.
  10. WCCs and C-sequences Definition (Todorcevic) Suppose that κ is a

    regular, uncountable cardinal. A C-sequence (over κ) is a sequence Cα | α < κ such that, for all limit α < κ, Cα is a club in α. The C-sequence is trivial if there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α ⊆ Cβ. Theorem (Todorcevic) Suppose that κ is a regular, uncountable cardinal. TFAE:
  11. WCCs and C-sequences Definition (Todorcevic) Suppose that κ is a

    regular, uncountable cardinal. A C-sequence (over κ) is a sequence Cα | α < κ such that, for all limit α < κ, Cα is a club in α. The C-sequence is trivial if there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α ⊆ Cβ. Theorem (Todorcevic) Suppose that κ is a regular, uncountable cardinal. TFAE: 1 κ is weakly compact;
  12. WCCs and C-sequences Definition (Todorcevic) Suppose that κ is a

    regular, uncountable cardinal. A C-sequence (over κ) is a sequence Cα | α < κ such that, for all limit α < κ, Cα is a club in α. The C-sequence is trivial if there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α ⊆ Cβ. Theorem (Todorcevic) Suppose that κ is a regular, uncountable cardinal. TFAE: 1 κ is weakly compact; 2 for every C-sequence Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α.
  13. WCCs and pair-colorings Definition Suppose that θ ≤ κ are

    cardinals. Then κ → (κ)2 θ is the assertion that,
  14. WCCs and pair-colorings Definition Suppose that θ ≤ κ are

    cardinals. Then κ → (κ)2 θ is the assertion that, for every c : [κ]2 → θ,
  15. WCCs and pair-colorings Definition Suppose that θ ≤ κ are

    cardinals. Then κ → (κ)2 θ is the assertion that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such that |c“[X]2| = 1.
  16. WCCs and pair-colorings Definition Suppose that θ ≤ κ are

    cardinals. Then κ → (κ)2 θ is the assertion that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such that |c“[X]2| = 1. By Ramsey’s theorem, ℵ0 → (ℵ0)2 n for all n < ω.
  17. WCCs and pair-colorings Definition Suppose that θ ≤ κ are

    cardinals. Then κ → (κ)2 θ is the assertion that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such that |c“[X]2| = 1. By Ramsey’s theorem, ℵ0 → (ℵ0)2 n for all n < ω. For regular, uncountable κ, the following are equivalent:
  18. WCCs and pair-colorings Definition Suppose that θ ≤ κ are

    cardinals. Then κ → (κ)2 θ is the assertion that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such that |c“[X]2| = 1. By Ramsey’s theorem, ℵ0 → (ℵ0)2 n for all n < ω. For regular, uncountable κ, the following are equivalent: 1 κ is weakly compact;
  19. WCCs and pair-colorings Definition Suppose that θ ≤ κ are

    cardinals. Then κ → (κ)2 θ is the assertion that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such that |c“[X]2| = 1. By Ramsey’s theorem, ℵ0 → (ℵ0)2 n for all n < ω. For regular, uncountable κ, the following are equivalent: 1 κ is weakly compact; 2 κ → (κ)2 2 ;
  20. WCCs and pair-colorings Definition Suppose that θ ≤ κ are

    cardinals. Then κ → (κ)2 θ is the assertion that, for every c : [κ]2 → θ, there is an unbounded X ⊆ κ such that |c“[X]2| = 1. By Ramsey’s theorem, ℵ0 → (ℵ0)2 n for all n < ω. For regular, uncountable κ, the following are equivalent: 1 κ is weakly compact; 2 κ → (κ)2 2 ; 3 κ → (κ)2 θ for all θ < κ.
  21. WCCs and chain conditions Definition For a property P and

    a cardinal θ, let us say that P is θ-productive if, whenever (Qi )i<θ all have P, then i<θ Qi also has P. (All products here are taken with full support.)
  22. WCCs and chain conditions Definition For a property P and

    a cardinal θ, let us say that P is θ-productive if, whenever (Qi )i<θ all have P, then i<θ Qi also has P. (All products here are taken with full support.) Definition Suppose that κ is a regular, uncountable cardinal and Q is a poset. Q is κ-Knaster if, whenever A ⊆ Q has size κ, there is B ⊆ A, also of size κ, consisting of pairwise compatible conditions.
  23. WCCs and chain conditions Definition For a property P and

    a cardinal θ, let us say that P is θ-productive if, whenever (Qi )i<θ all have P, then i<θ Qi also has P. (All products here are taken with full support.) Definition Suppose that κ is a regular, uncountable cardinal and Q is a poset. Q is κ-Knaster if, whenever A ⊆ Q has size κ, there is B ⊆ A, also of size κ, consisting of pairwise compatible conditions. If κ is weakly compact, then κ-Knaster = κ-c.c., and both properties are θ-productive for all θ < κ.
  24. I. The C-sequence number

  25. The C-sequence number Recall: κ is weakly compact iff for

    every C-sequence Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α.
  26. The C-sequence number Recall: κ is weakly compact iff for

    every C-sequence Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α. Definition Suppose that κ is a regular, uncountable cardinal. The C-sequence number, χ(κ), is defined as follows.
  27. The C-sequence number Recall: κ is weakly compact iff for

    every C-sequence Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α. Definition Suppose that κ is a regular, uncountable cardinal. The C-sequence number, χ(κ), is defined as follows. • If κ is weakly compact, then χ(κ) = 0.
  28. The C-sequence number Recall: κ is weakly compact iff for

    every C-sequence Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α. Definition Suppose that κ is a regular, uncountable cardinal. The C-sequence number, χ(κ), is defined as follows. • If κ is weakly compact, then χ(κ) = 0. • Otherwise, χ(κ) is the smallest χ such that,
  29. The C-sequence number Recall: κ is weakly compact iff for

    every C-sequence Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α. Definition Suppose that κ is a regular, uncountable cardinal. The C-sequence number, χ(κ), is defined as follows. • If κ is weakly compact, then χ(κ) = 0. • Otherwise, χ(κ) is the smallest χ such that, for every C-sequence Cα | α < κ ,
  30. The C-sequence number Recall: κ is weakly compact iff for

    every C-sequence Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α. Definition Suppose that κ is a regular, uncountable cardinal. The C-sequence number, χ(κ), is defined as follows. • If κ is weakly compact, then χ(κ) = 0. • Otherwise, χ(κ) is the smallest χ such that, for every C-sequence Cα | α < κ , there are an unbounded ∆ ⊆ κ and a function b : κ → [κ]χ such that,
  31. The C-sequence number Recall: κ is weakly compact iff for

    every C-sequence Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α. Definition Suppose that κ is a regular, uncountable cardinal. The C-sequence number, χ(κ), is defined as follows. • If κ is weakly compact, then χ(κ) = 0. • Otherwise, χ(κ) is the smallest χ such that, for every C-sequence Cα | α < κ , there are an unbounded ∆ ⊆ κ and a function b : κ → [κ]χ such that, for all α < κ, ∆ ∩ α ⊆ β∈b(α) Cβ.
  32. The C-sequence number Recall: κ is weakly compact iff for

    every C-sequence Cα | α < κ , there is an unbounded ∆ ⊆ κ such that, for every α < κ, there is β < κ such that ∆ ∩ α = Cβ ∩ α. Definition Suppose that κ is a regular, uncountable cardinal. The C-sequence number, χ(κ), is defined as follows. • If κ is weakly compact, then χ(κ) = 0. • Otherwise, χ(κ) is the smallest χ such that, for every C-sequence Cα | α < κ , there are an unbounded ∆ ⊆ κ and a function b : κ → [κ]χ such that, for all α < κ, ∆ ∩ α ⊆ β∈b(α) Cβ. Note that χ(κ) ∈ {0, 1} iff every C-sequence over κ is trivial.
  33. Some basic facts χ(κ) can be interpreted as a sort

    of measure of how far away κ is from being weakly compact. This view is bolstered by the following facts.
  34. Some basic facts χ(κ) can be interpreted as a sort

    of measure of how far away κ is from being weakly compact. This view is bolstered by the following facts. (Note that χ(κ) ≤ sup(κ ∩ Reg) for all κ.)
  35. Some basic facts χ(κ) can be interpreted as a sort

    of measure of how far away κ is from being weakly compact. This view is bolstered by the following facts. (Note that χ(κ) ≤ sup(κ ∩ Reg) for all κ.) • Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
  36. Some basic facts χ(κ) can be interpreted as a sort

    of measure of how far away κ is from being weakly compact. This view is bolstered by the following facts. (Note that χ(κ) ≤ sup(κ ∩ Reg) for all κ.) • Every stationary subset of κ ∩ cof(> χ(κ)) reflects. • If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0.
  37. Some basic facts χ(κ) can be interpreted as a sort

    of measure of how far away κ is from being weakly compact. This view is bolstered by the following facts. (Note that χ(κ) ≤ sup(κ ∩ Reg) for all κ.) • Every stationary subset of κ ∩ cof(> χ(κ)) reflects. • If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0. • (If χ(κ) > 1, then χ(κ) ≥ ℵ0.)
  38. Some basic facts χ(κ) can be interpreted as a sort

    of measure of how far away κ is from being weakly compact. This view is bolstered by the following facts. (Note that χ(κ) ≤ sup(κ ∩ Reg) for all κ.) • Every stationary subset of κ ∩ cof(> χ(κ)) reflects. • If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0. • (If χ(κ) > 1, then χ(κ) ≥ ℵ0.) • If (κ, < ℵ0) holds, then χ(κ) = sup(κ ∩ Reg).
  39. Inaccessible cardinals Theorem (LH-Rinot) Suppose that χ ≤ κ are

    cardinals such that κ is weakly compact and χ is either 1 or an infinite, regular cardinal. Then there is a forcing extension in which κ remains inaccessible and χ(κ) = χ.
  40. Inaccessible cardinals Theorem (LH-Rinot) Suppose that χ ≤ κ are

    cardinals such that κ is weakly compact and χ is either 1 or an infinite, regular cardinal. Then there is a forcing extension in which κ remains inaccessible and χ(κ) = χ. Sketch of proof. First, force with an Easton-support iteration P that adds a Cohen subset to every inaccessible α < κ. In V P, the weak compactness of κ is preserved by adding a Cohen subset to κ.
  41. Inaccessible cardinals Theorem (LH-Rinot) Suppose that χ ≤ κ are

    cardinals such that κ is weakly compact and χ is either 1 or an infinite, regular cardinal. Then there is a forcing extension in which κ remains inaccessible and χ(κ) = χ. Sketch of proof. First, force with an Easton-support iteration P that adds a Cohen subset to every inaccessible α < κ. In V P, the weak compactness of κ is preserved by adding a Cohen subset to κ. Define Q as follows, depending on the value of χ.
  42. Inaccessible cardinals Theorem (LH-Rinot) Suppose that χ ≤ κ are

    cardinals such that κ is weakly compact and χ is either 1 or an infinite, regular cardinal. Then there is a forcing extension in which κ remains inaccessible and χ(κ) = χ. Sketch of proof. First, force with an Easton-support iteration P that adds a Cohen subset to every inaccessible α < κ. In V P, the weak compactness of κ is preserved by adding a Cohen subset to κ. Define Q as follows, depending on the value of χ. • If χ = 1, let Q be Kunen’s forcing to add a homogeneous κ-Souslin tree.
  43. Inaccessible cardinals Sketch of proof, cont. • If ℵ0 ≤

    χ < κ, let Q be the forcing to add a ind(κ, χ)-sequence.
  44. Inaccessible cardinals Sketch of proof, cont. • If ℵ0 ≤

    χ < κ, let Q be the forcing to add a ind(κ, χ)-sequence. • If χ = κ, let Q be the forcing to add a non-reflecting stationary subset of κ.
  45. Inaccessible cardinals Sketch of proof, cont. • If ℵ0 ≤

    χ < κ, let Q be the forcing to add a ind(κ, χ)-sequence. • If χ = κ, let Q be the forcing to add a non-reflecting stationary subset of κ. V P∗ ˙ Q is the desired model. The key point is that, in all cases, the weak compactness of κ can be resurrected by a κ-distributive forcing.
  46. Inaccessible cardinals Sketch of proof, cont. • If ℵ0 ≤

    χ < κ, let Q be the forcing to add a ind(κ, χ)-sequence. • If χ = κ, let Q be the forcing to add a non-reflecting stationary subset of κ. V P∗ ˙ Q is the desired model. The key point is that, in all cases, the weak compactness of κ can be resurrected by a κ-distributive forcing. If χ = 1, this forcing is, moreover, κ-cc, which immediately gives χ(κ) = 1.
  47. Inaccessible cardinals Sketch of proof, cont. • If ℵ0 ≤

    χ < κ, let Q be the forcing to add a ind(κ, χ)-sequence. • If χ = κ, let Q be the forcing to add a non-reflecting stationary subset of κ. V P∗ ˙ Q is the desired model. The key point is that, in all cases, the weak compactness of κ can be resurrected by a κ-distributive forcing. If χ = 1, this forcing is, moreover, κ-cc, which immediately gives χ(κ) = 1. If χ = κ, χ(κ) = κ follows from the fact that every stationary subset of κ ∩ cof(> χ(κ)) reflects.
  48. Inaccessible cardinals Sketch of proof, cont. • If ℵ0 ≤

    χ < κ, let Q be the forcing to add a ind(κ, χ)-sequence. • If χ = κ, let Q be the forcing to add a non-reflecting stationary subset of κ. V P∗ ˙ Q is the desired model. The key point is that, in all cases, the weak compactness of κ can be resurrected by a κ-distributive forcing. If χ = 1, this forcing is, moreover, κ-cc, which immediately gives χ(κ) = 1. If χ = κ, χ(κ) = κ follows from the fact that every stationary subset of κ ∩ cof(> χ(κ)) reflects. If ℵ0 ≤ χ < κ, a slightly more delicate argument is needed.
  49. Successor cardinals If λ is an infinite cardinal, then, χ(λ+)

    ≤ λ.
  50. Successor cardinals If λ is an infinite cardinal, then, χ(λ+)

    ≤ λ. Moreover, since there is a C-sequence Cα | α < λ+ such that otp(Cα) ≤ λ for all α < λ+, an easy ordinal arithmetic argument yields cf(λ) ≤ χ(λ+).
  51. Successor cardinals If λ is an infinite cardinal, then, χ(λ+)

    ≤ λ. Moreover, since there is a C-sequence Cα | α < λ+ such that otp(Cα) ≤ λ for all α < λ+, an easy ordinal arithmetic argument yields cf(λ) ≤ χ(λ+). In particular, if λ is regular, then χ(λ+) = λ.
  52. Successor cardinals If λ is an infinite cardinal, then, χ(λ+)

    ≤ λ. Moreover, since there is a C-sequence Cα | α < λ+ such that otp(Cα) ≤ λ for all α < λ+, an easy ordinal arithmetic argument yields cf(λ) ≤ χ(λ+). In particular, if λ is regular, then χ(λ+) = λ. If λ is a singular cardinal, the story is more interesting.
  53. Successors of singular cardinals Theorem (LH-Rinot) Suppose that λ is

    a singular limit of supercompact cardinals. Then χ(λ+) = cf(λ).
  54. Successors of singular cardinals Theorem (LH-Rinot) Suppose that λ is

    a singular limit of supercompact cardinals. Then χ(λ+) = cf(λ). Moreover, if θ ∈ (cf(λ), λ) is a regular cardinal, then there is a cofinality-preserving forcing extension in which χ(λ+) = θ.
  55. Successors of singular cardinals Theorem (LH-Rinot) Suppose that λ is

    a singular limit of supercompact cardinals. Then χ(λ+) = cf(λ). Moreover, if θ ∈ (cf(λ), λ) is a regular cardinal, then there is a cofinality-preserving forcing extension in which χ(λ+) = θ. Theorem (LH-Rinot) Suppose that λ is a supercompact cardinal. In a Prikry extension using a normal measure over λ, we have χ(λ+) = ℵ0.
  56. Successors of singular cardinals Theorem (LH-Rinot) Suppose that λ is

    a singular limit of supercompact cardinals. Then χ(λ+) = cf(λ). Moreover, if θ ∈ (cf(λ), λ) is a regular cardinal, then there is a cofinality-preserving forcing extension in which χ(λ+) = θ. Theorem (LH-Rinot) Suppose that λ is a supercompact cardinal. In a Prikry extension using a normal measure over λ, we have χ(λ+) = ℵ0. Moreover, collapses can be interleaved into the Prikry forcing so that, in the generic extension, λ = ℵω and χ(ℵω+1) = ℵ0.
  57. II. Unbounded colorings

  58. U( ( (· · ·) ) ) Definition Suppose that

    µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such that,
  59. U( ( (· · ·) ) ) Definition Suppose that

    µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such that, for every χ < χ,
  60. U( ( (· · ·) ) ) Definition Suppose that

    µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such that, for every χ < χ, for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets,
  61. U( ( (· · ·) ) ) Definition Suppose that

    µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such that, for every χ < χ, for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every i < θ,
  62. U( ( (· · ·) ) ) Definition Suppose that

    µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such that, for every χ < χ, for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every i < θ, there is B ⊆ A of size µ such that,
  63. U( ( (· · ·) ) ) Definition Suppose that

    µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such that, for every χ < χ, for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every i < θ, there is B ⊆ A of size µ such that, for all distinct a, b ∈ B, we have min(c[a × b]) ≥ i.
  64. U( ( (· · ·) ) ) Definition Suppose that

    µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such that, for every χ < χ, for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every i < θ, there is B ⊆ A of size µ such that, for all distinct a, b ∈ B, we have min(c[a × b]) ≥ i. • U(κ, 2, θ, 2) is a strong negation of κ → (κ)2 θ .
  65. U( ( (· · ·) ) ) Definition Suppose that

    µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such that, for every χ < χ, for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every i < θ, there is B ⊆ A of size µ such that, for all distinct a, b ∈ B, we have min(c[a × b]) ≥ i. • U(κ, 2, θ, 2) is a strong negation of κ → (κ)2 θ . • If µ∗ ≥ µ and χ∗ ≥ χ, then U(κ, µ∗, θ, χ∗) implies U(κ, µ, θ, χ).
  66. U( ( (· · ·) ) ) Definition Suppose that

    µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such that, for every χ < χ, for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every i < θ, there is B ⊆ A of size µ such that, for all distinct a, b ∈ B, we have min(c[a × b]) ≥ i. • U(κ, 2, θ, 2) is a strong negation of κ → (κ)2 θ . • If µ∗ ≥ µ and χ∗ ≥ χ, then U(κ, µ∗, θ, χ∗) implies U(κ, µ, θ, χ). There is no such monotonicity in the third coordinate.
  67. U( ( (· · ·) ) ) Definition Suppose that

    µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such that, for every χ < χ, for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every i < θ, there is B ⊆ A of size µ such that, for all distinct a, b ∈ B, we have min(c[a × b]) ≥ i. • U(κ, 2, θ, 2) is a strong negation of κ → (κ)2 θ . • If µ∗ ≥ µ and χ∗ ≥ χ, then U(κ, µ∗, θ, χ∗) implies U(κ, µ, θ, χ). There is no such monotonicity in the third coordinate. • Instances of this principle are implicit in previous work of, for instance, Galvin, Todorcevic, and Shelah.
  68. A first construction Theorem (LH-Rinot) Suppose that θ, χ <

    κ are regular cardinals and there are a sequence Hi | i < θ of pairwise disjoint stationary subsets of κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅.
  69. A first construction Theorem (LH-Rinot) Suppose that θ, χ <

    κ are regular cardinals and there are a sequence Hi | i < θ of pairwise disjoint stationary subsets of κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then U(κ, κ, θ, χ) holds.
  70. A first construction Theorem (LH-Rinot) Suppose that θ, χ <

    κ are regular cardinals and there are a sequence Hi | i < θ of pairwise disjoint stationary subsets of κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then U(κ, κ, θ, χ) holds. Corollary Any of the following statements entails U(κ, κ, θ, χ).
  71. A first construction Theorem (LH-Rinot) Suppose that θ, χ <

    κ are regular cardinals and there are a sequence Hi | i < θ of pairwise disjoint stationary subsets of κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then U(κ, κ, θ, χ) holds. Corollary Any of the following statements entails U(κ, κ, θ, χ). 1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ).
  72. A first construction Theorem (LH-Rinot) Suppose that θ, χ <

    κ are regular cardinals and there are a sequence Hi | i < θ of pairwise disjoint stationary subsets of κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then U(κ, κ, θ, χ) holds. Corollary Any of the following statements entails U(κ, κ, θ, χ). 1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ). 2 κ = λ+ and λ is regular.
  73. A first construction Theorem (LH-Rinot) Suppose that θ, χ <

    κ are regular cardinals and there are a sequence Hi | i < θ of pairwise disjoint stationary subsets of κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then U(κ, κ, θ, χ) holds. Corollary Any of the following statements entails U(κ, κ, θ, χ). 1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ). 2 κ = λ+ and λ is regular. 3 κ = λ+ and cf(λ) = θ.
  74. A first construction Theorem (LH-Rinot) Suppose that θ, χ <

    κ are regular cardinals and there are a sequence Hi | i < θ of pairwise disjoint stationary subsets of κ ∩ cof(≥ χ) and a C-sequence Cα | α < κ such that, for all α < κ and all sufficiently large i < θ, Cα ∩ Hi = ∅. Then U(κ, κ, θ, χ) holds. Corollary Any of the following statements entails U(κ, κ, θ, χ). 1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ). 2 κ = λ+ and λ is regular. 3 κ = λ+ and cf(λ) = θ. 4 (κ) holds.
  75. Further results Theorem (LH-Rinot) Suppose that κ is an inaccessible

    cardinal, θ, χ < κ are regular cardinals, and S ⊆ κ is a stationary set such that either
  76. Further results Theorem (LH-Rinot) Suppose that κ is an inaccessible

    cardinal, θ, χ < κ are regular cardinals, and S ⊆ κ is a stationary set such that either 1 S ⊆ cof(χ) and ♣(S) holds; or
  77. Further results Theorem (LH-Rinot) Suppose that κ is an inaccessible

    cardinal, θ, χ < κ are regular cardinals, and S ⊆ κ is a stationary set such that either 1 S ⊆ cof(χ) and ♣(S) holds; or 2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal.
  78. Further results Theorem (LH-Rinot) Suppose that κ is an inaccessible

    cardinal, θ, χ < κ are regular cardinals, and S ⊆ κ is a stationary set such that either 1 S ⊆ cof(χ) and ♣(S) holds; or 2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal. Then U(κ, κ, θ, χ) holds.
  79. Further results Theorem (LH-Rinot) Suppose that κ is an inaccessible

    cardinal, θ, χ < κ are regular cardinals, and S ⊆ κ is a stationary set such that either 1 S ⊆ cof(χ) and ♣(S) holds; or 2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal. Then U(κ, κ, θ, χ) holds. Theorem (LH-Rinot) Suppose that λ is singular, θ < λ is regular, and either
  80. Further results Theorem (LH-Rinot) Suppose that κ is an inaccessible

    cardinal, θ, χ < κ are regular cardinals, and S ⊆ κ is a stationary set such that either 1 S ⊆ cof(χ) and ♣(S) holds; or 2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal. Then U(κ, κ, θ, χ) holds. Theorem (LH-Rinot) Suppose that λ is singular, θ < λ is regular, and either 1 2λ = λ+; or
  81. Further results Theorem (LH-Rinot) Suppose that κ is an inaccessible

    cardinal, θ, χ < κ are regular cardinals, and S ⊆ κ is a stationary set such that either 1 S ⊆ cof(χ) and ♣(S) holds; or 2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal. Then U(κ, κ, θ, χ) holds. Theorem (LH-Rinot) Suppose that λ is singular, θ < λ is regular, and either 1 2λ = λ+; or 2 there is a collection of fewer than cf(λ)-many stationary subsets of λ+ that do not reflect simultaneously.
  82. Further results Theorem (LH-Rinot) Suppose that κ is an inaccessible

    cardinal, θ, χ < κ are regular cardinals, and S ⊆ κ is a stationary set such that either 1 S ⊆ cof(χ) and ♣(S) holds; or 2 S ⊆ cof(≥ χ) and reflects at no inaccessible cardinal. Then U(κ, κ, θ, χ) holds. Theorem (LH-Rinot) Suppose that λ is singular, θ < λ is regular, and either 1 2λ = λ+; or 2 there is a collection of fewer than cf(λ)-many stationary subsets of λ+ that do not reflect simultaneously. Then U(λ+, λ+, θ, cf(λ)) holds.
  83. Consistency results Theorem (LH-Rinot) Suppose that κ is a weakly

    compact cardinal and θ < κ is regular. Then there is a cofinality-preserving forcing extension in which
  84. Consistency results Theorem (LH-Rinot) Suppose that κ is a weakly

    compact cardinal and θ < κ is regular. Then there is a cofinality-preserving forcing extension in which 1 U(κ, κ, θ, κ) holds;
  85. Consistency results Theorem (LH-Rinot) Suppose that κ is a weakly

    compact cardinal and θ < κ is regular. Then there is a cofinality-preserving forcing extension in which 1 U(κ, κ, θ, κ) holds; 2 U(κ, 2, θ , θ+) fails for all regular θ ∈ κ \ {θ}.
  86. Consistency results Theorem (LH-Rinot) Suppose that κ is a weakly

    compact cardinal and θ < κ is regular. Then there is a cofinality-preserving forcing extension in which 1 U(κ, κ, θ, κ) holds; 2 U(κ, 2, θ , θ+) fails for all regular θ ∈ κ \ {θ}. Theorem (LH-Rinot) Suppose that λ is a singular limit of supercompact cardinals.
  87. Consistency results Theorem (LH-Rinot) Suppose that κ is a weakly

    compact cardinal and θ < κ is regular. Then there is a cofinality-preserving forcing extension in which 1 U(κ, κ, θ, κ) holds; 2 U(κ, 2, θ , θ+) fails for all regular θ ∈ κ \ {θ}. Theorem (LH-Rinot) Suppose that λ is a singular limit of supercompact cardinals. 1 U(λ+, 2, θ, cf(λ)+) fails for every regular θ ∈ λ \ {cf(λ)}.
  88. Consistency results Theorem (LH-Rinot) Suppose that κ is a weakly

    compact cardinal and θ < κ is regular. Then there is a cofinality-preserving forcing extension in which 1 U(κ, κ, θ, κ) holds; 2 U(κ, 2, θ , θ+) fails for all regular θ ∈ κ \ {θ}. Theorem (LH-Rinot) Suppose that λ is a singular limit of supercompact cardinals. 1 U(λ+, 2, θ, cf(λ)+) fails for every regular θ ∈ λ \ {cf(λ)}. 2 Suppose that θ ∈ (cf(λ), λ) is regular. There is a cofinality-preserving forcing extension in which U(λ+, λ+, θ, λ+) holds but U(λ+, 2, θ , θ+) fails for all regular θ ∈ λ \ {θ, cf(λ)}.
  89. Prikry forcing Theorem (LH-Rinot) Suppose that λ is a supercompact

    cardinal. In the Prikry extension obtained using a normal measure over λ, U(λ+, 2, θ, ℵ1) fails for all regular, uncountable θ < λ.
  90. Prikry forcing Theorem (LH-Rinot) Suppose that λ is a supercompact

    cardinal. In the Prikry extension obtained using a normal measure over λ, U(λ+, 2, θ, ℵ1) fails for all regular, uncountable θ < λ. Moreover, one may interleave collapses in the Prikry forcing to obtain a model in which λ = ℵω and U(ℵω+1, 2, ℵn, ℵ1) fails for all 1 ≤ n < ω.
  91. III. Infinite productivity

  92. Productivity of Knasterness Recall that, if κ is a regular

    cardinal, then the κ-Knaster property is always finitely productive.
  93. Productivity of Knasterness Recall that, if κ is a regular

    cardinal, then the κ-Knaster property is always finitely productive. Moreover, if κ is weakly compact, then the κ-Knaster property is θ-productive for all θ < κ.
  94. Productivity of Knasterness Recall that, if κ is a regular

    cardinal, then the κ-Knaster property is always finitely productive. Moreover, if κ is weakly compact, then the κ-Knaster property is θ-productive for all θ < κ. This leads naturally to the following question: For what values of κ can the κ-Knaster property be even ℵ0-productive?
  95. Productivity of Knasterness Recall that, if κ is a regular

    cardinal, then the κ-Knaster property is always finitely productive. Moreover, if κ is weakly compact, then the κ-Knaster property is θ-productive for all θ < κ. This leads naturally to the following question: For what values of κ can the κ-Knaster property be even ℵ0-productive? In particular, is it consistent that there is a successor cardinal κ for which the κ-Knaster property is ℵ0-productive?
  96. Some previous results Theorem (Cox-L¨ ucke) If the existence of

    a weakly compact cardinal is consistent, then it is consistent that there is an inaccessible, non-weakly compact cardinal κ such that the κ-Knaster property is θ-productive for all θ < κ.
  97. Some previous results Theorem (Cox-L¨ ucke) If the existence of

    a weakly compact cardinal is consistent, then it is consistent that there is an inaccessible, non-weakly compact cardinal κ such that the κ-Knaster property is θ-productive for all θ < κ. Theorem (LH-L¨ ucke) Suppose that κ is a regular cardinal and the κ-Knaster property is ℵ0-productive. Then κ is weakly compact in L.
  98. IV. Connections

  99. The C-sequence number and unbounded colorings Theorem (LH-Rinot) Suppose that

    ℵ0 ≤ χ ≤ θ = cf(θ) < κ and there is a closed witness to U(κ, 2, θ, χ).
  100. The C-sequence number and unbounded colorings Theorem (LH-Rinot) Suppose that

    ℵ0 ≤ χ ≤ θ = cf(θ) < κ and there is a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ.
  101. The C-sequence number and unbounded colorings Theorem (LH-Rinot) Suppose that

    ℵ0 ≤ χ ≤ θ = cf(θ) < κ and there is a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ. Theorem (Todorcevic, LH-Rinot) There is a closed witness to U(κ, κ, ℵ0, χ(κ)).
  102. The C-sequence number and unbounded colorings Theorem (LH-Rinot) Suppose that

    ℵ0 ≤ χ ≤ θ = cf(θ) < κ and there is a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ. Theorem (Todorcevic, LH-Rinot) There is a closed witness to U(κ, κ, ℵ0, χ(κ)). Theorem (LH-Rinot) There is a closed witness to U(κ, κ, χ(κ), χ(κ)).
  103. The C-sequence number and unbounded colorings Theorem (LH-Rinot) Suppose that

    ℵ0 ≤ χ ≤ θ = cf(θ) < κ and there is a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ. Theorem (Todorcevic, LH-Rinot) There is a closed witness to U(κ, κ, ℵ0, χ(κ)). Theorem (LH-Rinot) There is a closed witness to U(κ, κ, χ(κ), χ(κ)). Conjecture (LH-Rinot) For every regular θ < κ, U(κ, κ, θ, χ(κ)) holds.
  104. Unbounded colorings and productivity Lemma (LH-Rinot) Suppose that θ ≤

    χ < κ are infinite, regular cardinals, κ is (< χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that
  105. Unbounded colorings and productivity Lemma (LH-Rinot) Suppose that θ ≤

    χ < κ are infinite, regular cardinals, κ is (< χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that • P is well-met and χ-directed closed with greatest lower bounds;
  106. Unbounded colorings and productivity Lemma (LH-Rinot) Suppose that θ ≤

    χ < κ are infinite, regular cardinals, κ is (< χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that • P is well-met and χ-directed closed with greatest lower bounds; • Pτ is κ-Knaster for all τ < θ;
  107. Unbounded colorings and productivity Lemma (LH-Rinot) Suppose that θ ≤

    χ < κ are infinite, regular cardinals, κ is (< χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that • P is well-met and χ-directed closed with greatest lower bounds; • Pτ is κ-Knaster for all τ < θ; • Pθ is not κ-c.c.
  108. Unbounded colorings and productivity Lemma (LH-Rinot) Suppose that θ ≤

    χ < κ are infinite, regular cardinals, κ is (< χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that • P is well-met and χ-directed closed with greatest lower bounds; • Pτ is κ-Knaster for all τ < θ; • Pθ is not κ-c.c. Corollary For every infinite λ, the λ+-Knaster property is not ℵ0-productive.
  109. Productivity, cont. Sketch of proof of corollary. Recall that cf(λ)

    ≤ χ(λ+) and that U(λ+, λ+, ℵ0, χ(λ+)) holds.
  110. Productivity, cont. Sketch of proof of corollary. Recall that cf(λ)

    ≤ χ(λ+) and that U(λ+, λ+, ℵ0, χ(λ+)) holds. Now apply the Lemma with θ = χ = ℵ0 (every infinite cardinal is (< ℵ0)-inaccessible.)
  111. Productivity, cont. Sketch of proof of corollary. Recall that cf(λ)

    ≤ χ(λ+) and that U(λ+, λ+, ℵ0, χ(λ+)) holds. Now apply the Lemma with θ = χ = ℵ0 (every infinite cardinal is (< ℵ0)-inaccessible.) Conjecture Suppose κ is a regular, uncountable cardinal. Then the κ-Knaster property is ℵ0-productive if and only if χ(κ) < ℵ0, i.e., if and only if every C-sequence over κ is trivial.
  112. 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!
  113. References The work in this talk comes from the following

    two papers, both jointly written with Assaf Rinot: • Knaster and friends I: Closed colorings and precalibers. Preprint available at math.biu.ac.il/∼lambiec/knaster1.pdf • Knaster and friends II: Subadditive colorings and the C-sequence number. Coming soon! All feedback is welcome! Artwork: “Bands of Color in Four Directions” by Sol Lewitt
  114. Thank you!