Adding many Cohen reals

Transcript

1. Adding many Cohen reals Chris Lambie-Hanson Institute of Mathematics Czech Academy of Sciences European Set Theory Conference 2022
2. This talk includes joint work with Jeffrey Bergfalk, Radek Honzik, Michael Hruˇ s´ ak, ˇ S´ arka Stejskalov´ a, and Andy Zucker.
3. I. Introduction
4. A guiding theme This talk is centered around the following two (related) central themes: 1 the combinatorics of the real line; 2 compactness principles, e.g., stationary reflection, tree properties, failures of square, etc. In particular, we will be interested in questions that can be interpreted as asking, “to what extent does adding reals preserve instances of compactness?” We will be especially, but not exclusively, interested in Cohen reals.
5. Why? There are various reasons to be interested in such questions, including: 1 investigating the effect of various set-theoretic hypotheses on the continuum; 2 proving new consistency results about the reals or related structures; 3 establishing consistency results about structures unrelated to the reals, but for which adding reals is nonetheless useful.
6. II. Preserving compactness
7. Guessing models Definition (Viale–Weiß) Let θ ≥ ℵ2 be a regular cardinal and M ≺ H(θ). Given a set x ∈ M and a subset d ⊆ x, we say that • d is M-approximated if, for every z ∈ M ∩ Pω1 (x), we have d ∩ z ∈ M. • d is M-guessed if there is e ∈ M such that d ∩ M = e ∩ M. M is a guessing model if |M| = ω1 ⊆ M and, for all x ∈ M and all d ⊆ x, if d is M-approximated, then d is M-guessed. GMP is the assertion that, for every regular θ ≥ ℵ2, the set of guessing models is stationary in Pω2 H(θ). GMP (also known as ISP(ω2)) is a consequence of the Proper Forcing Axiom, and also entails many of the consequences of PFA, including the Singular Cardinals Hypothesis (SCH) and failures of square.
8. Indestructibility of GMP Cox and Krueger proved that, unlike PFA, GMP places no limitations on 2ℵ0 other than 2ℵ0 > ℵ1. In particular, they produced a special model of GMP in which it is indestructible under adding any number of Cohen reals. In recent work, we showed that this in fact holds in any model of GMP. Theorem (Honzik–LH–Stejskalov´ a, ’2X, [3]) Suppose that P is the forcing to add some number of Cohen reals and M is a guessing model with P ∈ M. Then MP is a guessing model in V P. In particular, if GMP holds, then it continues to hold after adding any number of Cohen reals.
9. 2ℵ0 vs. 2ℵ1 We saw that GMP places no upper bound on the size of the continuum, and, unlike MA, does not imply 2ℵ1 = 2ℵ0 , since GMP is compatible with cf(2ℵ0 ) = ℵ1. However, we were able to show that GMP nonetheless entails that 2ℵ1 is as small as possible relative to 2ℵ0 . Theorem (LH–Stejskalov´ a, ’2X, [5]) Suppose that GMP holds. Then 2ℵ1 = 2ℵ0 if cf(2ℵ0 ) ̸= ℵ1 (2ℵ0 )+ if cf(2ℵ0 ) = ℵ1.
10. The Suslin Hypothesis There are some principles that adding Cohen reals will never preserve. For example, recall that a Suslin tree is a tree of height ω1 with no uncountable chains or antichains. The Suslin Hypothesis (SH) is the assertion that there are no Suslin trees. By a result of Shelah, adding Cohen reals will never preserve SH, since adding even a single Cohen real adds a Suslin tree. Therefore, if we want to preserve SH, we need a different technique.
11. Special trees Recall that a tree T of height ω1 is special if there is a function f : T → ω that is injective on chains, i.e., if u ̸= v ∈ T and f (u) = f (v), then u and v are incomparable in T. It is immediate that a special tree cannot have an uncountable branch and cannot be Suslin. MAℵ1 implies that all trees of height and size ℵ1 without cofinal branches are special. Definition (Baumgartner) A tree T of height ω1 is B-special if there is a function f : T → ω such that, for all u, v, w ∈ T, if u ≤T v, w and f (u) = f (v) = f (w), then v and w are T-comparable.
12. Special trees Recall that a tree T of height ω1 is special if there is a function f : T → ω that is injective on chains, i.e., if u ̸= v ∈ T and f (u) = f (v), then u and v are incomparable in T. It is immediate that a special tree cannot have an uncountable branch and cannot be Suslin. MAℵ1 implies that all trees of height and size ℵ1 without cofinal branches are special. Definition (Baumgartner) A tree T of height ω1 is B-special if there is a function f : T → ω such that, for all u, v, w ∈ T, if u ≤T v, w and f (u) = f (v) = f (w), then v and w are T-comparable.
13. B-special trees If T is a tree of height ω1 with no uncountable branch, then T is B-special if and only if it is special. If T is a tree of height and size ω1 and T is B-special, then • T has at most ω1-many cofinal branches; • T does not gain any new cofinal branches in any outer model with the same ω1. In particular, if every tree of height and size ω1 is B-special, then there are no weak Kurepa trees. If PFA holds, then every tree of height and size ω1 is B-special.
14. Adding random reals Theorem (Laver ’87) If any number of random reals is added to a model of MAω1 , then, in the extension, every tree of size and height ω1 with no uncountable branches is special. This was significant in part because it completed the proof that Suslin’s Hypothesis places no limitations on the value of the continuum. Theorem (LH-Stejskalov´ a, ‘2X, [5]) If any number of random reals is added to a model of PFA, then, in the extension, every tree of size and height ω1 is B-special.
15. Indestructible guessing models Question Is GMP preserved after adding random reals? What about just over models of PFA? One reason for interest in this question is the following: By a result of Cox and Krueger, if every tree of height and size ω1 is B-special, then every guessing model in fact remains a guessing model in any outer model with the same ω1. This indestructible version of GMP is known to be compatible with arbitrarily large values of the continuum, but is not known to be compatible with cf(2ℵ0 ) = ℵ1.
16. III. Partition relations
17. Classical partition relations Recall the Hungarian notation for partition relations: If λ, µ, and ν are cardinals and k is a natural number, then ν → (µ)k λ is the assertion that, for every function c : [ν]k → λ, there is H ⊆ ν of size µ such that c ↾ [H]k is constant. • The infinite Ramsey theorem states that, for all k, m < ω, ℵ0 → (ℵ0)k m . • An uncountable cardinal κ is weakly compact if and only if κ → (κ)2 2 .
18. Counterexamples at 2ℵ0 There are two simple, very strong counterexamples to natural generalizations of Ramsey’s theorem to 2ℵ0 . • Define ∆ : [ω2]2 → ω by letting ∆(f , g) be the least j such that f (j) ̸= g(j). Then ∆ witnesses 2ℵ0 ̸→ (3)2 ℵ0 . • Define d : [R]2 → 2 as follows. Fix a well-ordering ≺ on R and let d(x, y) = 0 if ≺ agrees with the usual ordering of R on the order of x and y, and d(x, y) = 1 otherwise. Then d witnesses 2ℵ0 ̸→ (ℵ1)2 2 .
19. Positive results at 2ℵ0 Theorem Either of the following implies that 2ℵ0 → (2ℵ0 , α)2 for all α < ω1 : 1 (Kunen ’71) 2ℵ0 is real-valued measurable; 2 (Todorcevic ’86) V is an extension of a ground model V0 by the forcing to add weakly-compact-many Cohen (or Sacks, Silver, random, etc.) reals. Theorem (Zhang ’20) V Add(ω,ℶω) |= R →+ (ℵ0 )d for all d < ω, i.e., for all f : R → d, there is an infinite X ⊆ R such that f ↾ X + X is constant. This built on prior work of Komj´ ath, Leader, Russell, Shelah, Soukup, and Vidny´ anszky. Theorem (Hindman–Leader–Strauss ’17) If 2ℵ0 < ℵω , then there is d < ω for which R ̸→+ (ℵ0 )d .
20. Connectedness The relation ν → (µ)2 λ can be phrased in graph-theoretic language: Whenever the edges of the complete graph on ν-many vertices are colored with λ-many colors, we can find a complete monochromatic subgraph of size µ. In search of nontrivial partition relations that can hold at small uncountable cardinals, one might try to slightly weaken the requirement that the monochromatic subgraph we obtain is complete. One natural way to approach this is via considerations of connectedness. Definition Let G = (V, E) be a graph. 1 G is connected if, for all u, v ∈ V, there are u0, u1, . . . , un ∈ V such that u0 = u, un = v, and, for all i < n, {ui , ui+1} ∈ E. 2 G is κ-connected if it is connected and remains connected after removing any fewer than κ-many vertices.
21. Highly connectedness Definition (Bergfalk-Hruˇ s´ ak-Shelah ‘20) 1 A graph G is highly connected if it is |G|-connected. 2 The partition relation ν →hc (µ)2 λ is the assertion that, for every c : [ν]2 → λ, there are H ⊆ ν of size µ and a highly connected graph (H, E) such that c ↾ E is constant. Note: A finite graph is highly connected if and only if it is complete, so the relation ν →hc (µ)2 λ can be seen as a genuine generalization of the classical finite Ramsey partition relations.
22. Earlier results Some immediate negative results in ZFC: for all infinite λ, • for all µ ≥ 3, λ+ ̸→hc (µ)2 λ ; • 2λ ̸→hc (2λ)2 λ . Theorem (Bergfalk-Hruˇ s´ ak-Shelah ‘20) If the existence of a weakly compact cardinal is consistent, then it is consistent that 2ℵ1 →hc (2ℵ1 )2 ω . Theorem (Hruˇ s´ ak-Shelah-Zhang ‘2X) If the existence of a measurable cardinal is consistent, then it is consistent that ω2 →hc (ω2)2 ω .
23. Square bracket relations To state our results, we need a variation on our highly connected partition relations. Definition The partition ν →hc [µ]2 λ,κ (resp. ν → [µ]2 λ,<κ ) is the assertion that, for every coloring c : [ν]2 → λ, there is H ⊆ ν of size µ and a highly connected graph (H, E) such that |c“E| ≤ κ (resp. |c“E| < κ). In other words, ν →hc [µ]2 λ,κ says that whenever the edges of the complete graph on ν are colored with λ-many colors, one can find a highly connected subgraph of size µ on which only κ-many colors are obtained.
24. An equiconsistency result Theorem (LH ‘22 [4]) Suppose that ν is a weakly compact cardinal and P is the poset to add ν-many Cohen reals. Then, in V P, for all λ < ν we have 2ℵ0 →hc [2ℵ0 ]2 λ,2 . This is sharp, since we know that • 2ℵ0 ̸→hc (2ℵ0 )2 ω ; • 2ℵ0 ̸→ [ℵ0]2 ℵ0,<ℵ0 Theorem (LH ‘22 [4]) The following are equiconsistent over ZFC: 1 There exists a weakly compact cardinal. 2 2ℵ0 →hc [2ℵ0 ]2 ω,2 .
25. Polish spaces Definition Let d ≥ 1. The principle PGd (for Polish grid) is the following assertion: whenever X0, . . . , Xd−1 are perfect Polish spaces, k < ω, and c : i<d Xi → k, there are somewhere dense subsets Y0 ⊆ X0, . . . , Yd−1 ⊆ Xd−1 such that c ↾ i<d Yi is constant. PGd (ℵ0) is defined in the same way, except considering functions c : i<d Xi → ω. This partition relation is of interest in its own right, and also in the context of the Halpern-L¨ auchli theorem and variations thereof; for example, if PGd holds, it provides a very quick proof of the d-dimensional Halpern-L¨ auchli theorem.
26. Sharp consistency results Theorem (LH-Zucker, 2X) Let 2 ≤ d < ω. 1 If PGd holds, then 2ℵ0 ≥ ℵd−1. 2 If PGd (ℵ0) holds, then 2ℵ0 ≥ ℵd . 3 If χ ≥ (ℶd−1)+, then V Add(ω,χ) |= PGd (ℵ0). Corollary For all 2 ≤ d < ω, PGd (ℵ0) is incompatible 2ℵ0 < ℵd but is compatible with 2ℵ0 = ℵd . We note that some similar results were obtained independently by N. Stefanovi´ c.
27. Aligned sets We now give a vague idea of how these results are proved. In effect, they proceed by showing that adding Cohen reals preserves a sufficient fragment of either (1) the weak compactness of a cardinal in the ground model or (2) a ground model instance of the Erd˝ os-Rado theorem. Definition Suppose that a and b are finite sets of ordinals. We say that a and b are aligned if |a| = |b| and, for all γ ∈ a ∩ b, we have |a ∩ γ| = |b ∩ γ|. 0 1 2 3 4 5 a b c a and b are aligned; b and c are aligned; a and c are not aligned.
28. Higher Delta-systems Lemma Suppose that 1 ≤ d < ω, χ ≥ ℶ+ d−1 , and P is the forcing to add any number of Cohen reals. Let ⟨pa | a ∈ [χ]d ⟩ be a sequence of elements from P, and let f : [χ]d → ω be a function. Then there is H ∈ [χ]ω1 such that • f ↾ [H]d is constant; • for all a, b ∈ [H]d , if a and b are aligned, then pa and pb are compatible (and more). If χ is weakly compact, then we can in fact find H ∈ [χ]χ as above.
29. Sketch of proof We will give some ideas of the proof of the fact that, if χ ≥ (ℶd−1)+, then V Add(ω,χ) |= PGd (ℵ0). Let P = Add(ω, χ), p ∈ P, and ˙ c a name for a function from (ωω)d → ω. For α < χ, let ˙ fα be a name for the αth Cohen real added by P. For each a = {α0, . . . , αd−1}< in [χ]d , find some condition qa ≤ p deciding the value of ˙ c( ˙ fα0 , . . . , ˙ fαd−1 ), say as na < ω. Apply the previous lemma to find some H ∈ [χ]ω1 and n < ω such that • na = n for all a ∈ [H]d ; • qa and qb are compatible for all aligned a, b ∈ [H]d (and more). Let H0 be the first ω-many elements of H, H1 the next ω-many elements, etc., up to Hd−1.
30. Sketch of proof (cont.) Let G be a P-generic filter containing the “root” of the d-dimensional system ⟨qa | a ∈ [H]d ⟩. Working in V [G], using genericity, the uniformity of ⟨qa | a ∈ [H]d ⟩, and appropriate bookkeeping, recursively construct sets ⟨A0, . . . , Ad−1⟩ such that • for each i < d, Ai is an infinite subset of Hi ; • for each i < d, {fα | α ∈ Ai } is somewhere dense in ωω; • for all a ∈ [H]d such that |a ∩ Ai | = 1 for all i < d, we have qa ∈ G. For each i < d, let Yi = {fα | α ∈ Ai }. Then c ↾ i<d Yi is constant, taking value n.
31. Some questions We have a sharp result for PGd (ℵ0), but a 1-cardinal gap for PGd : • PGd implies 2ℵ0 ≥ ℵd−1; • PGd (ℵ0), and hence PGd , is compatible with 2ℵ0 = ℵd . Our forcing argument is not sensitive enough to distinguish between finitely many and countably many colors. Question Is PGd compatible with 2ℵ0 = ℵd−1? In particular, is PG2 compatible with CH? Is PG2 in fact a theorem of ZFC? Proposition (LH-Zucker, ’2X, [6]) PG2 is true in ZFC when restricted to 2-colorings, i.e., if X0 and X1 are perfect Polish spaces and c : X0 × X1 → 2, then there are somewhere dense sets Y0 ⊆ X0 and Y1 ⊆ X1 such that c ↾ (Y0 × Y1) is constant.
32. IV. Vanishing derived limits
33. Some notation Given f , g ∈ ωω, let f ∧ g denote their greatest lower bound in (ωω, ≤), and let I(f ) = {(k, m) ∈ ω × ω | m ≤ f (k)}.
34. Nontrivial coherent families Given f , g ∈ ωω, let f ∧ g denote their greatest lower bound in (ωω, ≤), and let I(f ) = {(k, m) ∈ ω × ω | m ≤ f (k)}. Definition Let Φ = ⟨φf : I(f ) → Z | f ∈ ωω⟩ be a family of functions. 1 Φ is coherent if φf ↾ I(f ∧ g) =∗ φg ↾ I(f ∧ g) for all f , g ∈ ωω. 2 Φ is trivial if there is a function ψ : ω × ω → Z such that φf =∗ ψ ↾ I(f ) for all f ∈ ωω. Clearly, a trivial family is coherent. A nontrivial coherent family Φ is a clear example of set theoretic incompactness: each local family ⟨φf | f < g⟩ (for a fixed g ∈ ωω) is trivial, as witnessed by φg itself, but the entire family is not.
35. 2-dimensional nontrivial coherence Definition Let Φ = ⟨φfg : I(f ∧ g) → Z | f , g ∈ ωω⟩. 1 Φ is alternating if φfg = −φgf for all f , g ∈ ωω. 2 Φ is 2-coherent if it is alternating and φfg + φgh =∗ φfh for all f , g, h ∈ ωω. (All functions restricted to I(f ∧ g ∧ h).) 3 Φ is 2-trivial if there is a family Ψ = ⟨ψf : I(f ) → Z | f ∈ ωω⟩ such that ψg − ψf =∗ φfg for all f , g ∈ ωω. Again, a 2-trivial family is 2-coherent. A non-2-trivial 2-coherent family Φ is an example of incompactness: each local family ⟨φfg | f , g < h⟩ (for a fixed h ∈ ωω) is 2-trivial, as witnessed by the family ⟨−φfh | f < h⟩, but the entire family is not.
36. n-dimensional nontrivial coherence Given a sequence ⃗ f = (f0, . . . , fn−1) and i < n, ⃗ f i is the sequence of length n − 1 formed by removing fi from ⃗ f . Definition Fix n ≥ 2, and let Φ = φ⃗ f : I(∧⃗ f ) → Z ⃗ f ∈ (ωω)n . 1 Φ is alternating if φ⃗ f = sgn(σ)φ σ(⃗ f ) for all ⃗ f ∈ (ωω)n and all permutations σ. 2 Φ is n-coherent if it is alternating and n i=0 (−1)i φ⃗ f i =∗ 0 for all ⃗ f ∈ (ωω)n+1. 3 Φ is n-trivial if there is an alternating family ψ⃗ f : I(∧⃗ f ) → Z ⃗ f ∈ (ωω)n−1 such that n−1 i=0 (−1)i ψ⃗ f i =∗ φ⃗ f for all ⃗ f ∈ (ωω)n.
37. The system A A is an inverse system of abelian groups, indexed by (ωω, ≤), defined as follows. A = ⟨Af , πfg | f ≤ g ∈ ωω⟩, where • Af = I(f ) Z • πfg : Ag → Af is the natural projection map. In other words, Ag is the group of finitely supported functions φ : I(g) → Z, and, if f ≤ g, then πfg takes such a function φ to φ ↾ I(f ). A and its derived inverse limits limn A occur naturally in a variety of contexts, including the study of strong homology and condensed mathematics.
38. limn A and nontrivial coherence Coherent and trivial can now be thought of as 1-coherent and 1-trivial. Theorem (Mardeˇ si´ c-Prasolov (n = 1), Bergfalk (n ≥ 2)) Fix n ≥ 1. Then limn A = 0 if and only if every n-coherent family Φ = φ⃗ f ⃗ f ∈ (ωω)n is n-trivial. Thus, to prove that limn A = 0, it suffices to show that every n-coherent family is n-trivial.
39. lim1 A Theorem (Mardeˇ si´ c-Prasolov, Simon, ’88) CH ⇒ lim1 A ̸= 0. Theorem (Dow-Simon-Vaughan, ‘89) d = ℵ1 ⇒ lim1 A ̸= 0. PFA ⇒ lim1 A = 0. Theorem (Todorcevic, ‘98) OCA ⇒ lim1 A = 0. MAℵ1 ̸⇒ lim1 A = 0. Theorem (Kamo, ‘93) V Add(ω,ω2) |= “ lim1 A = 0”.
40. Higher limits The higher derived limits of A remained mysterious until a few years ago. In particular, it remained open whether it was consistent that limn A = 0 for all n ≥ 1. Theorem (Bergfalk, ‘17) PFA ⇒ lim2 A ̸= 0. Theorem (Bergfalk-LH, ‘21, [2]) Suppose that κ is a weakly compact cardinal and that P is a length-κ finite support iteration of Hechler forcings. Then V P |= “∀n ≥ 1 limn A = 0”. Theorem (Bergfalk-Hruˇ s´ ak-LH, ’22, [1]) V Add(ω, ℶω) |= “∀n ≥ 1 limn A = 0”.
41. Some notes on proofs • The proofs of the previous two theorems make use of the higher-dimensional ∆-systems from the previous section to show that, for any name for an n-coherent family, we can find a name for a “large” subfamily of it that is forced to be trivial. • We then argue that the triviality of this large subfamily can be propagated to the entire family. • The proof in the weakly compact Hechler model is easier, since the Hechler reals form a <∗-increasing, cofinal subset of ωω, and the “large subfamily” of the first point can be taken to be indexed by a cofinal subset thereof. • Without a weakly compact cardinal, the “large subfamily” is necessarily of size less than 2ℵ0 in the extension. Since any subset of Hechler reals of size less than 2ℵ0 is <∗-bounded, Hechler reals are not useful in this context. Due to their flexibility, Cohen reals work, but the fact that the resulting order structure of (ωω, <∗) in the generic extension is much more complicated than in the Hechler model makes the proof more difficult.
42. Some questions Question If 2ℵ0 < ℵω, must there be 1 ≤ n < ω for which limn A ̸= 0? Question There are natural generalizations Aκ,λ of A, where κ and λ are infinite cardinals and A = Aℵ0,ℵ0 . What can be said about limn Aκ,λ in general?
43. V. Nonvanishing derived limits
44. More coherence and triviality We can also make sense of coherent and trivial families indexed by ordinals rather than members of ωω. Definition Let δ be an ordinal and A an abelian group. A family Φ = ⟨φα : α → A | α < δ} is 1 coherent if φα =∗ φβ ↾ α for all α < β < δ; 2 trivial if there is a function ψ : δ → A such that φα =∗ ψ ↾ α for all α < δ. If δ is a regular uncountable cardinal and |A| < δ, then a nontrivial coherent family ⟨φα : α → A | α < δ⟩ is essentially a coherent δ-Aronszajn tree.
45. Higher-dimensional coherence Definition Let n ≥ 2, δ an ordinal, and A an abelian group. A family Φ = ⟨φa : min(a) → A | a ∈ [δ]n⟩ is 1 n-coherent if, for all b ∈ [δ]n+1, n i=0 (−1)i φbi =∗ 0; 2 n-trivial if there exists a family ⟨ψa : min(a) → A | a ∈ [δ]n−1⟩ such that, for all a ∈ [δ]n, n−1 i=0 (−1)i ψai =∗ φa.
46. Some facts • If δ is an ordinal, n ≥ 1 and A is an abelian group, then the existence of a nontrivial n-coherent family ⟨φa : min(a) → A | a ∈ [δ]n⟩ is equivalent to the nonvanishing of the ˇ Cech cohomology group ˇ Hn(δ, A). • (Goblot ’70) If cf(δ) < ℵn, then every n-coherent family ⟨φa | a ∈ [δ]n⟩ is trivial. • (Bergfalk ’21) For each n ≥ 2, there is a nontrivial n-coherent family Φ = φa : min(a) → ωn Z a ∈ [ωn]n . So nontrivial n-coherence isolates behavior that provably first occurs at ℵn.
47. Getting smaller groups Such a nontrivial n-coherent family Φ = φa : min(a) → ωn Z a ∈ [ωn]n can be seen as an n-dimensional analogue of a “wide” coherent Aronszajn tree. In search of a true n-dimensional analogue of coherent Aronszajn trees, we would want nontrivial n-coherent families mapping into smaller abelian groups (ideally Z or Z/2Z). It’s tempting to try to turn a Φ as above into such a family Φ′ by simply adding up the entries in each row, i.e., φ′ a (ξ) = η∈sppt(φa(ξ)) φa(ξ)(η). This maintains n-coherence, but in general there seems to be no reason why it would maintain nontriviality.
48. Adding Cohen reals Theorem (Bergfalk–LH, ’2X) Fix n ≥ 2, and let χ ≥ ℵn. In V Add(ω,χ), there is a nontrivial, n-coherent family Φ′ = ⟨φ′ a : min(a) → Z | a ∈ [ωn]n⟩. The proof proceeds by fixing a carefully chosen nontrivial n-coherent family Φ = ⟨φa : min(a) → ωn Z | a ∈ [ωn]n⟩ in the ground model. In the forcing extension, let h : χ → ω be the generic object. Define Φ′ by letting, for all a ∈ [ωn]n and all ξ < min(a), φ′ a (ξ) = η∈sppt(φa(ξ)) h(η) · φa(ξ)(η).
49. Some questions Question Is it true in ZFC that, for all n ≥ 1, there is a nontrivial n-coherent family Φ′ = ⟨φ′ a : min(a) → Z | a ∈ [ωn ]n⟩? The proof of the previous theorem, when applied to the n = 1 case, yields a coherent Suslin tree, by an argument of Todorcevic. This raises the natural question as to whether there is a meaningful higher-dimensional generalization of Suslin-ness. Question Is there a natural notion of Suslin-ness for n-coherent families of functions? If so, does it apply to the families constructed in the proof of the previous theorem? Question Is it consistent that every 2-coherent family ⟨φa : min(a) → A | a ∈ [ω3 ]2⟩ is trivial?
50. References Jeffrey Bergfalk, Michael Hruˇ s´ ak, and Chris Lambie-Hanson, Simultaneously vanishing higher derived limits without large cardinals, J. Math. Log. (2022), To appear. Jeffrey Bergfalk and Chris Lambie-Hanson, Simultaneously vanishing higher derived limits, Forum Math. Pi 9 (2021), Paper No. e4, 31. Radek Honzik, Chris Lambie-Hanson, and ˇ S´ arka Stejskalov´ a, Indestructibility of some compactness principles over models of PFA, (2022), Preprint. Chris Lambie-Hanson, A note on highly connected and well-connected Ramsey theory, Fund. Math. (2022), To appear. Chris Lambie-Hanson and ˇ S´ arka Stejskalov´ a, Strong tree properties and cardinal arithmetic, (2022), In preparation. Chris Lambie-Hanson and Andy Zucker, Polish space partition principles and the Halpern-L¨ auchli theorem, (2022), In preparation.
51. Thank you!