Chris Lambie-Hanson
January 13, 2018

# Squares, ascent paths, and chain conditions

Using a variety of square principles, we obtain results on the consistency strengths of the non-existence of kappa-Aronszajn trees with narrow ascent paths and of the infinite productivity of strong kappa-chain conditions. In particular, we show that, if kappa is an uncountable regular cardinal that is not weakly compact in L, then:
1. for every lambda < kappa, there is a kappa-Aronszajn tree with a lambda-ascent path;
2. there is a kappa-Knaster poset P such that P^omega does not have the kappa-chain condition;
3. there is a kappa-Knaster poset that is not kappa-stationarily layered.
This answers questions of Cox and Lücke and consists of joint work with Philipp Lücke.

January 13, 2018

## Transcript

1. ### Squares, ascent paths, and chain conditions (Joint work with Philipp

L¨ ucke) Chris Lambie-Hanson Department of Mathematics Bar-Ilan University Joint Mathematics Meetings AMS-ASL Special Session on Set Theory, Logic and Ramsey Theory San Diego, CA 11 January 2018

3. ### Special Trees Deﬁnition Suppose κ is a regular cardinal and

(T, <T ) is a tree of height κ.
4. ### Special Trees Deﬁnition Suppose κ is a regular cardinal and

(T, <T ) is a tree of height κ. 1 (T, <T ) is a κ-tree if all of its levels have size < κ.
5. ### Special Trees Deﬁnition Suppose κ is a regular cardinal and

(T, <T ) is a tree of height κ. 1 (T, <T ) is a κ-tree if all of its levels have size < κ. 2 (T, <T ) is a κ-Aronszajn tree if it is a κ-tree and has no chains of length κ.
6. ### Special Trees Deﬁnition Suppose κ is a regular cardinal and

(T, <T ) is a tree of height κ. 1 (T, <T ) is a κ-tree if all of its levels have size < κ. 2 (T, <T ) is a κ-Aronszajn tree if it is a κ-tree and has no chains of length κ. 3 If κ = µ+ for some cardinal µ, then (T, <T ) is special if it can be decomposed into µ antichains.
7. ### Special Trees Deﬁnition Suppose κ is a regular cardinal and

(T, <T ) is a tree of height κ. 1 (T, <T ) is a κ-tree if all of its levels have size < κ. 2 (T, <T ) is a κ-Aronszajn tree if it is a κ-tree and has no chains of length κ. 3 If κ = µ+ for some cardinal µ, then (T, <T ) is special if it can be decomposed into µ antichains. Remark This deﬁnition has been generalized by Todorcevic to apply also to trees of inaccessible height. Special κ-trees are easily seen to be κ-Aronszajn trees. Their specialness provides a concrete obstacle to the existence of coﬁnal branches that persists in outer models in which κ remains regular.
8. ### Ascent Paths Deﬁnition Suppose λ < κ are regular cardinals

and (T, <T ) is a tree of height κ. A λ-ascent path through (T, <T ) is a sequence bα : λ → Tα | α < κ such that, for all α < β < κ and all suﬃciently large i < λ, we have bα(i) <T bβ(i).
9. ### Ascent Paths Deﬁnition Suppose λ < κ are regular cardinals

and (T, <T ) is a tree of height κ. A λ-ascent path through (T, <T ) is a sequence bα : λ → Tα | α < κ such that, for all α < β < κ and all suﬃciently large i < λ, we have bα(i) <T bβ(i). A λ-ascent path through (T, <T ) can be seen as a “fuzzy coﬁnal branch,” or as a coﬁnal branch through the ultraproduct of (T, <T ) by a uniform ultraﬁlter over λ.
10. ### Ascent Paths Theorem (Shelah, Todorcevic) Suppose that λ < κ

are regular cardinals, κ is not the successor of a cardinal of coﬁnality λ, and (T, <T ) is a special tree of height κ. Then there is no λ-ascent path through (T, <T ).
11. ### Ascent Paths Theorem (Shelah, Todorcevic) Suppose that λ < κ

are regular cardinals, κ is not the successor of a cardinal of coﬁnality λ, and (T, <T ) is a special tree of height κ. Then there is no λ-ascent path through (T, <T ). Remark If µ is a singular cardinal and µ holds, then there is a special µ+-tree with a cf(µ)-ascent path.
12. ### Ascent Paths Theorem (Shelah, Todorcevic) Suppose that λ < κ

are regular cardinals, κ is not the successor of a cardinal of coﬁnality λ, and (T, <T ) is a special tree of height κ. Then there is no λ-ascent path through (T, <T ). Remark If µ is a singular cardinal and µ holds, then there is a special µ+-tree with a cf(µ)-ascent path. Question Under what conditions do κ-Aronszajn trees with λ-ascent paths exist?
13. ### Chain Conditions Deﬁnition Let κ be a regular cardinal, and

let P be a poset.
14. ### Chain Conditions Deﬁnition Let κ be a regular cardinal, and

let P be a poset. 1 P has the κ-chain condition (κ-c.c.) if every antichain of P has size < κ.
15. ### Chain Conditions Deﬁnition Let κ be a regular cardinal, and

let P be a poset. 1 P has the κ-chain condition (κ-c.c.) if every antichain of P has size < κ. 2 P is κ-Knaster if, whenever pα | α < κ is a sequence of conditions from P, there is an unbounded A ⊆ κ such that pα | α ∈ A consists of pairwise-compatible conditions.
16. ### Chain Conditions Deﬁnition Let κ be a regular cardinal, and

let P be a poset. 1 P has the κ-chain condition (κ-c.c.) if every antichain of P has size < κ. 2 P is κ-Knaster if, whenever pα | α < κ is a sequence of conditions from P, there is an unbounded A ⊆ κ such that pα | α ∈ A consists of pairwise-compatible conditions. 3 P is κ-stationarily layered if the collection of regular suborders of P is stationary in Pκ(P).
17. ### Chain Conditions Deﬁnition Let κ be a regular cardinal, and

let P be a poset. 1 P has the κ-chain condition (κ-c.c.) if every antichain of P has size < κ. 2 P is κ-Knaster if, whenever pα | α < κ is a sequence of conditions from P, there is an unbounded A ⊆ κ such that pα | α ∈ A consists of pairwise-compatible conditions. 3 P is κ-stationarily layered if the collection of regular suborders of P is stationary in Pκ(P). It is not hard to show that κ-stationarily layered ⇒ κ-Knaster ⇒ κ-c.c.
18. ### Productivity of Chain Conditions Questions about the productivity κ-c.c. have

spawned a great deal of research in set theory. Here, we look at the productivity of its strengthenings.
19. ### Productivity of Chain Conditions Questions about the productivity κ-c.c. have

spawned a great deal of research in set theory. Here, we look at the productivity of its strengthenings. Easily, the κ-Knaster property is always productive, i.e., if P and Q are κ-Knaster, then P × Q is κ-Knaster.
20. ### Productivity of Chain Conditions Questions about the productivity κ-c.c. have

spawned a great deal of research in set theory. Here, we look at the productivity of its strengthenings. Easily, the κ-Knaster property is always productive, i.e., if P and Q are κ-Knaster, then P × Q is κ-Knaster. If κ is weakly compact, then the κ-Knaster property is θ-productive for every θ < κ, i.e., whenever {Pα | α < θ} is a collection of κ-Knaster posets, then the full-support product α<θ Pα is κ-Knaster.
21. ### Productivity of Chain Conditions Questions about the productivity κ-c.c. have

spawned a great deal of research in set theory. Here, we look at the productivity of its strengthenings. Easily, the κ-Knaster property is always productive, i.e., if P and Q are κ-Knaster, then P × Q is κ-Knaster. If κ is weakly compact, then the κ-Knaster property is θ-productive for every θ < κ, i.e., whenever {Pα | α < θ} is a collection of κ-Knaster posets, then the full-support product α<θ Pα is κ-Knaster. Question Under what conditions is the κ-Knaster property inﬁnitely productive?
22. ### Strong Failures of Ramsey’s Theorem Both questions involve strong failures

to higher analogues of Ramsey’s theorem.
23. ### Strong Failures of Ramsey’s Theorem Both questions involve strong failures

to higher analogues of Ramsey’s theorem. Namely, if κ → (κ)2 λ , then:
24. ### Strong Failures of Ramsey’s Theorem Both questions involve strong failures

to higher analogues of Ramsey’s theorem. Namely, if κ → (κ)2 λ , then: • Every κ-tree with a λ-ascent path has a chain of length κ.
25. ### Strong Failures of Ramsey’s Theorem Both questions involve strong failures

to higher analogues of Ramsey’s theorem. Namely, if κ → (κ)2 λ , then: • Every κ-tree with a λ-ascent path has a chain of length κ. • If {Pi | i < λ} are posets and i<λ Pi is not κ-Knaster, then one of the Pi ’s is not κ-c.c.

27. ### (κ, λ) Deﬁnition Suppose 0 < λ < κ are

cardinals, with κ regular. (κ, λ) is the assertion that there is a sequence Cα | α ∈ acc(κ) such that:
28. ### (κ, λ) Deﬁnition Suppose 0 < λ < κ are

cardinals, with κ regular. (κ, λ) is the assertion that there is a sequence Cα | α ∈ acc(κ) such that: 1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that 0 < |Cα| ≤ λ;
29. ### (κ, λ) Deﬁnition Suppose 0 < λ < κ are

cardinals, with κ regular. (κ, λ) is the assertion that there is a sequence Cα | α ∈ acc(κ) such that: 1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that 0 < |Cα| ≤ λ; 2 for all α < β in acc(κ) and all C ∈ Cβ, if α ∈ acc(C), then C ∩ α ∈ Cα;
30. ### (κ, λ) Deﬁnition Suppose 0 < λ < κ are

cardinals, with κ regular. (κ, λ) is the assertion that there is a sequence Cα | α ∈ acc(κ) such that: 1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that 0 < |Cα| ≤ λ; 2 for all α < β in acc(κ) and all C ∈ Cβ, if α ∈ acc(C), then C ∩ α ∈ Cα; 3 there is no club D in κ such that, for all α ∈ acc(D), D ∩ α ∈ Cα.
31. ### (κ, λ) Deﬁnition Suppose 0 < λ < κ are

cardinals, with κ regular. (κ, λ) is the assertion that there is a sequence Cα | α ∈ acc(κ) such that: 1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that 0 < |Cα| ≤ λ; 2 for all α < β in acc(κ) and all C ∈ Cβ, if α ∈ acc(C), then C ∩ α ∈ Cα; 3 there is no club D in κ such that, for all α ∈ acc(D), D ∩ α ∈ Cα. (κ, 1) is more commonly denoted by (κ).
32. ### (κ, λ) Deﬁnition Suppose 0 < λ < κ are

cardinals, with κ regular. (κ, λ) is the assertion that there is a sequence Cα | α ∈ acc(κ) such that: 1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that 0 < |Cα| ≤ λ; 2 for all α < β in acc(κ) and all C ∈ Cβ, if α ∈ acc(C), then C ∩ α ∈ Cα; 3 there is no club D in κ such that, for all α ∈ acc(D), D ∩ α ∈ Cα. (κ, 1) is more commonly denoted by (κ). (κ, λ) can be seen as an instance of incompactness at κ.
33. ### (κ, λ) Deﬁnition Suppose 0 < λ < κ are

cardinals, with κ regular. (κ, λ) is the assertion that there is a sequence Cα | α ∈ acc(κ) such that: 1 for all α ∈ acc(κ), Cα is a collection of clubs in α such that 0 < |Cα| ≤ λ; 2 for all α < β in acc(κ) and all C ∈ Cβ, if α ∈ acc(C), then C ∩ α ∈ Cα; 3 there is no club D in κ such that, for all α ∈ acc(D), D ∩ α ∈ Cα. (κ, 1) is more commonly denoted by (κ). (κ, λ) can be seen as an instance of incompactness at κ. If κ is a regular cardinal and (κ) fails, then κ is weakly compact in L.
34. ### ind(κ, λ) Deﬁnition Suppose that λ < κ are inﬁnite,

regular cardinals. ind(κ, λ) is the assertion that there is a sequence Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that:
35. ### ind(κ, λ) Deﬁnition Suppose that λ < κ are inﬁnite,

regular cardinals. ind(κ, λ) is the assertion that there is a sequence Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that: 1 for all α ∈ acc(κ) and all i(α) ≤ i < λ, Cα,i is club in α;
36. ### ind(κ, λ) Deﬁnition Suppose that λ < κ are inﬁnite,

regular cardinals. ind(κ, λ) is the assertion that there is a sequence Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that: 1 for all α ∈ acc(κ) and all i(α) ≤ i < λ, Cα,i is club in α; 2 for all α ∈ acc(κ) and all i(α) ≤ i < j < λ, Cα,i ⊆ Cα,j ;
37. ### ind(κ, λ) Deﬁnition Suppose that λ < κ are inﬁnite,

regular cardinals. ind(κ, λ) is the assertion that there is a sequence Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that: 1 for all α ∈ acc(κ) and all i(α) ≤ i < λ, Cα,i is club in α; 2 for all α ∈ acc(κ) and all i(α) ≤ i < j < λ, Cα,i ⊆ Cα,j ; 3 for all α < β in acc(κ) and all i(β) ≤ i < λ, if α ∈ acc(Cβ,i ), then i(α) ≤ i and Cβ,i ∩ α = Cα,i ;
38. ### ind(κ, λ) Deﬁnition Suppose that λ < κ are inﬁnite,

regular cardinals. ind(κ, λ) is the assertion that there is a sequence Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that: 1 for all α ∈ acc(κ) and all i(α) ≤ i < λ, Cα,i is club in α; 2 for all α ∈ acc(κ) and all i(α) ≤ i < j < λ, Cα,i ⊆ Cα,j ; 3 for all α < β in acc(κ) and all i(β) ≤ i < λ, if α ∈ acc(Cβ,i ), then i(α) ≤ i and Cβ,i ∩ α = Cα,i ; 4 for all α < β in acc(κ), there is i(β) ≤ i < λ such that α ∈ acc(Cβ,i );
39. ### ind(κ, λ) Deﬁnition Suppose that λ < κ are inﬁnite,

regular cardinals. ind(κ, λ) is the assertion that there is a sequence Cα,i | α ∈ acc(κ), i(α) ≤ i < λ such that: 1 for all α ∈ acc(κ) and all i(α) ≤ i < λ, Cα,i is club in α; 2 for all α ∈ acc(κ) and all i(α) ≤ i < j < λ, Cα,i ⊆ Cα,j ; 3 for all α < β in acc(κ) and all i(β) ≤ i < λ, if α ∈ acc(Cβ,i ), then i(α) ≤ i and Cβ,i ∩ α = Cα,i ; 4 for all α < β in acc(κ), there is i(β) ≤ i < λ such that α ∈ acc(Cβ,i ); 5 there is no club D in κ and i < λ such that, for all α ∈ acc(D), we have i(α) ≤ i and D ∩ α = Cα,i .

41. ### Implications Clearly, ind(κ, λ) ⇒ (κ, λ). Theorem (LH-L¨ ucke)

If λ < κ are inﬁnite, regular cardinals and (κ) holds, then ind(κ, λ) holds.
42. ### Implications Clearly, ind(κ, λ) ⇒ (κ, λ). Theorem (LH-L¨ ucke)

If λ < κ are inﬁnite, regular cardinals and (κ) holds, then ind(κ, λ) holds. Theorem (LH, Shani) Assuming the consistency of large cardinals, it is consistent that (κ, 2) holds but ind(κ, λ) fails for all regular λ such that λ+ < κ.
43. ### Implications Clearly, ind(κ, λ) ⇒ (κ, λ). Theorem (LH-L¨ ucke)

If λ < κ are inﬁnite, regular cardinals and (κ) holds, then ind(κ, λ) holds. Theorem (LH, Shani) Assuming the consistency of large cardinals, it is consistent that (κ, 2) holds but ind(κ, λ) fails for all regular λ such that λ+ < κ. Theorem (Hayut-LH) Assuming the consistency of large cardinals, it is consistent that ind(κ, λ) holds but (κ, θ) fails for all θ < λ.

45. ### Ascent paths from squares Theorem (L¨ ucke) Suppose that λ

< κ are regular cardinals, S ⊆ Sκ λ is stationary, and there is a (κ)-sequence Cα | α ∈ acc(κ) such that acc(Cα) ∩ S = ∅ for all α ∈ acc(κ). Then there is a κ-Aronszajn tree with a λ-ascent path.
46. ### Ascent paths from squares Theorem (L¨ ucke) Suppose that λ

< κ are regular cardinals, S ⊆ Sκ λ is stationary, and there is a (κ)-sequence Cα | α ∈ acc(κ) such that acc(Cα) ∩ S = ∅ for all α ∈ acc(κ). Then there is a κ-Aronszajn tree with a λ-ascent path. Theorem (LH-L¨ ucke) Suppose that λ < κ are regular cardinals and ind(κ, λ) holds. Then there is a κ-Aronszajn tree with a λ-ascent path.
47. ### Ascent paths from squares Theorem (L¨ ucke) Suppose that λ

< κ are regular cardinals, S ⊆ Sκ λ is stationary, and there is a (κ)-sequence Cα | α ∈ acc(κ) such that acc(Cα) ∩ S = ∅ for all α ∈ acc(κ). Then there is a κ-Aronszajn tree with a λ-ascent path. Theorem (LH-L¨ ucke) Suppose that λ < κ are regular cardinals and ind(κ, λ) holds. Then there is a κ-Aronszajn tree with a λ-ascent path. Corollary (LH-L¨ ucke) Suppose that λ < κ are regular cardinals and (κ) holds. Then there is a κ-Aronszajn tree with a λ-ascent path.
48. ### Tree properties Recall that, if κ is a regular cardinal,

then the tree property at κ asserts that every κ-tree has a chain of length κ.
49. ### Tree properties Recall that, if κ is a regular cardinal,

then the tree property at κ asserts that every κ-tree has a chain of length κ. For a regular λ < κ, one can formulate a similar property by asserting that every κ-tree has a λ-ascent path.
50. ### Tree properties Recall that, if κ is a regular cardinal,

then the tree property at κ asserts that every κ-tree has a chain of length κ. For a regular λ < κ, one can formulate a similar property by asserting that every κ-tree has a λ-ascent path. Clearly, this property is implied by the tree property; a natural question is whether it is strictly weaker than the tree property.
51. ### Tree properties Recall that, if κ is a regular cardinal,

then the tree property at κ asserts that every κ-tree has a chain of length κ. For a regular λ < κ, one can formulate a similar property by asserting that every κ-tree has a λ-ascent path. Clearly, this property is implied by the tree property; a natural question is whether it is strictly weaker than the tree property. Theorem (LH-L¨ ucke) The following are equiconsistent over ZFC:
52. ### Tree properties Recall that, if κ is a regular cardinal,

then the tree property at κ asserts that every κ-tree has a chain of length κ. For a regular λ < κ, one can formulate a similar property by asserting that every κ-tree has a λ-ascent path. Clearly, this property is implied by the tree property; a natural question is whether it is strictly weaker than the tree property. Theorem (LH-L¨ ucke) The following are equiconsistent over ZFC: 1 there is a weakly compact cardinal;
53. ### Tree properties Recall that, if κ is a regular cardinal,

then the tree property at κ asserts that every κ-tree has a chain of length κ. For a regular λ < κ, one can formulate a similar property by asserting that every κ-tree has a λ-ascent path. Clearly, this property is implied by the tree property; a natural question is whether it is strictly weaker than the tree property. Theorem (LH-L¨ ucke) The following are equiconsistent over ZFC: 1 there is a weakly compact cardinal; 2 there are ℵ2-Aronszajn trees, but every ℵ2-Aronszajn tree has an ℵ0-ascent path.

55. ### Failure of Inﬁnite Productivity Theorem (LH-L¨ ucke) Suppose κ is

a regular, uncountable cardinal and ind(κ, ℵ0) holds. Then there is are posets {Pi | i < ω} such that: 1 each Pi is κ-Knaster; 2 i<ω Pi is not κ-c.c.
56. ### Failure of Inﬁnite Productivity Theorem (LH-L¨ ucke) Suppose κ is

a regular, uncountable cardinal and ind(κ, ℵ0) holds. Then there is are posets {Pi | i < ω} such that: 1 each Pi is κ-Knaster; 2 i<ω Pi is not κ-c.c. Proof Sketch. Let Cα,i | α ∈ acc(κ), i(α) ≤ i < ω be a ind(κ, ℵ0)-sequence. We can arrange so that, for all i < ω, {α ∈ acc(κ) | i(α) = i} is stationary.
57. ### Failure of Inﬁnite Productivity Theorem (LH-L¨ ucke) Suppose κ is

a regular, uncountable cardinal and ind(κ, ℵ0) holds. Then there is are posets {Pi | i < ω} such that: 1 each Pi is κ-Knaster; 2 i<ω Pi is not κ-c.c. Proof Sketch. Let Cα,i | α ∈ acc(κ), i(α) ≤ i < ω be a ind(κ, ℵ0)-sequence. We can arrange so that, for all i < ω, {α ∈ acc(κ) | i(α) = i} is stationary. For each i < ω, let Ti be the tree whose underlying set is {α ∈ acc(κ) | i(α) ≤ i}, ordered by α <Ti β iﬀ α ∈ acc(Cβ,i ).
58. ### Proof Sketch (Cont.) Let Pi be the poset consisting of

ﬁnite partial functions from Ti to ω that are injective on <Ti -chains, ordered by reverse inclusion. One can show that Pi is κ-Knaster.
59. ### Proof Sketch (Cont.) Let Pi be the poset consisting of

ﬁnite partial functions from Ti to ω that are injective on <Ti -chains, ordered by reverse inclusion. One can show that Pi is κ-Knaster. To show that i<ω Pi is not κ-c.c., consider the set A = {fα | α ∈ acc(κ)}, where fα ∈ i<ω Pi is deﬁned by fα(i) = ∅ if i < i(α) {(α, 0)} if i(α) ≤ i < ω
60. ### Proof Sketch (Cont.) Let Pi be the poset consisting of

ﬁnite partial functions from Ti to ω that are injective on <Ti -chains, ordered by reverse inclusion. One can show that Pi is κ-Knaster. To show that i<ω Pi is not κ-c.c., consider the set A = {fα | α ∈ acc(κ)}, where fα ∈ i<ω Pi is deﬁned by fα(i) = ∅ if i < i(α) {(α, 0)} if i(α) ≤ i < ω Now, if α < β are in acc(κ) and i(β) ≤ i < ω is such that α ∈ acc(Cβ,i ), then fα(i) and fβ(i) are incompatible in Pi , so fα and fβ are incompatible in i<ω Pi , and A is thus an antichain.
61. ### More Recent Results Theorem (LH-L¨ ucke) If κ is a

regular cardinal and ind(κ, ℵ0) holds, then there is a κ-Knaster poset that is not κ-stationarily layered.
62. ### More Recent Results Theorem (LH-L¨ ucke) If κ is a

regular cardinal and ind(κ, ℵ0) holds, then there is a κ-Knaster poset that is not κ-stationarily layered. Theorem (Hayut-LH) If κ is a regular cardinal and ind(κ, ℵ0) holds, then there are posets {Pi | i < ω} such that each Pi is κ-stationarily layered and i<ω Pi is not κ-c.c.
63. ### More Recent Results Theorem (LH-L¨ ucke) If κ is a

regular cardinal and ind(κ, ℵ0) holds, then there is a κ-Knaster poset that is not κ-stationarily layered. Theorem (Hayut-LH) If κ is a regular cardinal and ind(κ, ℵ0) holds, then there are posets {Pi | i < ω} such that each Pi is κ-stationarily layered and i<ω Pi is not κ-c.c. Theorem (LH-Rinot) Suppose that κ is a successor cardinal. Then there are posets {Pi | i < ω} such that each Pi is κ-Knaster and i<ω Pi is not κ-c.c.