SQUARE WITH BUILT-IN DIAMOND-PLUS

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1 SQUARE WITH BUILT-IN DIAMOND-PLUS ASSAF RINOT AND RALF SCHINDLER Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of the diamond principle, and prove that the strongest amongst them holds in L for every infinite cardinal. As an application, we prove that the following two hold in L: (1) For every infinite regular cardinal λ, there exists a special λ + -Aronszajn tree whose projection is almost Souslin; (2) For every infinite cardinal λ, there exists a respecting λ + -Kurepa tree; Roughly speaking, this means that this λ + -Kurepa tree looks very much like the λ + -Souslin trees that Jensen constructed in L. 1. Introduction In his seminal paper [13], Jensen initiated the study of the fine structure of Gödel s constructible universe, L, and proved that in this model, for every uncountable cardinal κ which is not weakly compact, there exists a κ-souslin tree. These fine-structural-constructions of Souslin trees were then factored through the combinatorial principles and (also due to Jensen), making the construction accessible to a wider audience of mathematicians. In [9], Gray introduced a combinatorial principle that forms a strong combination of and, which he denoted by. This principle turned out to be very fruitful, and, for instance, has recently been used to answer an old question of Hajnal in infinite graph theory [17]. In this paper, we introduce principles that combine with stronger forms of, such as * and +. We study the implication between these principles, and prove that the strongest amongst them, λ, holds in L for every infinite cardinal λ. As and have countless applications in infinite combinatorics, we expect the principles of this paper to prove fruitful, and allow deeper applications of the nature of L, outside of L. In this paper, we demonstrate the utility of the new principles by presenting applications to the theory of trees. It is proved: (1) * λ + λ<λ = λ entails the existence of a λ -respecting special λ + -Aronszajn tree whose projection is almost Souslin (to be defined below); (2) + λ entails the existence of a λ-respecting λ + -Kurepa tree; (3) λ entails the existence of a λ-respecting λ + -Kurepa tree with the additional feature of having no λ + -Aronszajn subtrees. Before giving the definition of a respecting tree, we first motivate it by briefly recalling how Jensen constructed a λ + -Souslin tree from the hypothesis that λ (E) + (E) holds for some stationary subset E λ +. 1 Date: December 7, Mathematics Subject Classification. Primary 03E45. Secondary 03E05. Key words and phrases. diamond principle, square principle, constructibility, walks on ordinals, Kurepa tree, almost Souslin tree, parameterized proxy principle. 1 The full details may be found in, e.g., [6, Theorem IV.2.4] or [18, Lemma 11.68]. 1

2 2 ASSAF RINOT AND RALF SCHINDLER Let C α α < λ + and S α α E witness the hypothesis. The construction of the tree T is by recursion, where at stage α < λ +, the α th -level, T α, is constructed. We start with T 0 a singleton, and for T α+1, we simply make sure that any node of T α admits two incompatible extensions in T α+1. The heart of the matter is the definition of T α for α limit nonzero, once T α = β<α T β has already been constructed. Here, for every node x T α, one identifies a canonical branch b α x which is cofinal in T α and goes through x. Of course, to be able to construct such a branch, we need to make sure that the process of climbing up through the levels of T α is always successful, i.e., that we never get stuck when trying to take a limit. For this, we advise with the -sequence, ensuring in advance that if α acc(c α ), then b α x would make an initial segment of b α x. 2 But we also need to seal antichains! For this, we advise with the -sequence, to decide whether T α should be equal to {b α x x T α}, or only to some carefully-chosen subset of it. Here is a possible abstraction of the above process. Definition 1.1. Suppose that T is a downward-closed family of functions from ordinals to some fixed set Ω, so that (T, ) forms a λ + -tree. Denote T X = {t T dom(t) X}. We say that T is λ -respecting if there exists a stationary subset E λ +, and a sequence of mappings b α : T C α α Ω { } α < λ + such that: (1) C α α < λ + is a λ (E)-sequence; (2) T α Im(b α ) for every α E; (3) if α acc(c α ) and x T C α, then b α (x) = b α (x) α. In particular, for every α E, any node y T α is essentially the limit of some canonical branch b α (x) for some x T C α. While working on their paper, the authors of [2] were considering the problem of constructing, say, an ℵ 3 -Souslin tree whose reduced ℵ 0 -power is ℵ 3 -Aronszajn, and whose reduced ℵ 1 -power is ℵ 3 - Kurepa. They realized that such a tree may be constructed, provided that there exists a respecting ℵ 3 -Kurepa tree. Question. Can a λ + -Kurepa tree be λ -respecting? At a first glance, this sounds unlikely, as λ + -Kurepa trees are usually obtained in a top-down fashion (one outright identifies λ ++ many whole functions from λ + to 2, and then verifies that the number of traces on any α < λ + is rather small), while respecting trees are described in a bottomup langauge. However, in this paper, we shall demonstrate that + λ allows the construction of such a λ + -Kurepa tree. In fact, we shall construct a λ + -Kurepa tree satisfying a considerably stronger form of λ -respecting, that is, λ (λ + )-respecting (see Definition 4.4 below). In another front, we shall prove that * λ entails the existence of a λ-sequence C for which the λ + -Aronszajn trees derived from the process of walks on ordinals [23] along C are respecting. This is also somewhat surprising, as the standard description of these trees is also top-down. However, the fact that these trees could be λ -respecting is already hinted in [21, Equation (*) on p. 267], based on a concept affine to Definition 1.1 and implicit in the proof of [20, Theorem 4.1]. Of course, the derived trees obtained here will be moreover λ (λ + )-respecting. For λ regular uncountable, we shall also ensure that the derived tree T (ρ 1 ) (which is a projection of the special tree T (ρ 0 )) is almost Souslin. 3 As for λ = ℵ 0, in [22], it was proved that Cohen 2 Here, acc(a) = {α < sup(a) sup(a α) = α > 0}. 3 The relevant definitions may be found on Section 4 below.

3 SQUARE WITH BUILT-IN DIAMOND-PLUS 3 modification of a C-sequence on ω 1 makes its T (ρ 1 ) almost Souslin. In [10], a similar result was obtained from * (ω 1 ). Altogether, we conclude that in L, for every infinite regular cardinal λ, there exists a special λ + -Aronszajn tree whose projection is almost Souslin Organization of this paper. In Section 2, we recall the definition of the principle λ, introduce the principles * λ, + λ, λ, and discuss the interrelations between them. In Section 3, we prove that if V = L, then λ holds for every infinite cardinal λ. In Section 4, we use the new principles to derive new types of λ + -trees. 2. Hybrid squares and diamonds In [9], Gray introduced the principle λ for λ a regular uncountable cardinal. In [1, S2], the definition was generalized to cover the case of λ singular. Then, in [3], the principle was generalized to cover the case λ = ℵ 0, as well. The outcome is as follows: Definition 2.1 ([9],[1],[3]). λ asserts the existence of (C α, S α ) α < λ + such that: (1) C α is a club in α of order-type (ω λ); (2) S α α; (3) if α acc(c α ), then (a) C α = C α α; (b) S α = S α α; (4) for every X λ + and every club D λ +, there exists a limit α < λ + with otp(acc(c α )) = λ, such that S α = X α and acc(c α ) D. Note that S α α < λ + forms a (E λ+ cf(λ) )-sequence, and if λ is uncountable, then C α α < λ + forms a λ -sequence. By [3], ω is equivalent to (ω 1 ). By [19], λ + (λ + ) does not imply λ for λ regular uncountable. By [16], λ + (λ + ) is equivalent to λ for every singular cardinal λ. Definition 2.2. * λ asserts the existence of (C α, X α, f α ) α < λ + such that: (1) C α is a club in α of order-type λ; (2) X α is a subset of P(α), of size λ; (3) f α : C α X α is a function, and a surjection whenever otp(c α ) = λ; (4) if α acc(c α ), then (a) C α = C α α; (b) f α (β) = f α (β) α for all β C α ; (5) for every subset X λ + and a club C λ +, the set { α < λ + C α * C & X α X α } contains a club; 4 (6) {α < λ + otp(c α ) = λ} is stationary in λ +. Of course, X α α < λ + forms a * (λ + )-sequence, and C α α < λ + forms a strong clubguessing sequence in the sense of [8]. In particular, by [12], * ω is actually stronger than * (ω 1 ). Definition λ asserts the existence of (C α, N α, f α ) α < λ + such that: (1) C α is a club in α of order-type λ; (2) N α is a rud-closed transitive set, λ = N α N α, with {f β β < α} {α} N α ; (3) f α : C α P(α) N α is a function; (4) if α acc(c α ), then 4 Here, A * B stands for the assertion that sup(a B) < sup(a).

4 4 ASSAF RINOT AND RALF SCHINDLER (a) C α = C α α; (b) f α (β) = f α (β) α for all β C α ; (5) for every subset X λ + and a club C λ +, there exists a club D λ + such that for all α D: C α * C; X α, D α N α ; if otp(c α ) = λ, then X α Im(f α ); (6) {α < λ + f α is surjective} is stationary in λ + ; (7) {N α α < λ + } is an increasing -chain converging to H λ +. Note that P (α) N α α < λ + forms a + (λ + )-sequence. As before, the result of [12] entails that + ω is stronger than + (ω 1 ). Definition 2.4. λ asserts the existence of (C α, N α, f α ) α < λ + such that Clauses (1) (7) of Definition 2.3 hold, with Clause (6) strengthened to (6) for every n, m < ω, end every Π n m-sentence η valid in a structure (λ +,, A i i < ω ), 5 there are stationarily many α < λ + for which all of the following hold: f α is surjective; A i α i < ω N α ; N α = η is valid in (α,, A i α i < ω ). Note that if (C α, N α, f α ) α < λ + witnesses λ, then N α α < λ + is not far from being a witness to Devlin s notion of a (λ + )-sequence (see [5]). Specifically, there are two differences: In (λ + ), the models N α are required to be p.r.-closed, while here they are only rud-closed; In (λ + ), the reflection property is restricted to Π 1 2-sentences, while here there is no restriction on the complexity of the sentences. Lemma 2.5. * λ entails the existence of a * λ -sequence (C α, X α, f α ) α < λ + with the additional property that for every club D λ +, there exists some limit α < λ + with otp(c α ) = λ such that C α D. Proof. Let (C α, X α, f α ) α < λ + be a witness to * λ. Fix ς : λ λ such that for all i < λ, ς(i) i and ς 1 {i} is cofinal in λ. Let α < λ + be arbitrary. For all j < λ, denote C j α = {γ C α otp(c α γ) j}. Let π α : otp(c α ) C α denote the monotone enumeration of C α. Define f α : C α X α by stipulating f α(β) = f α (π α (ς(πα 1 (β))). It is easy to see that if α acc(c α ), then π α = π α otp(c α ), and hence f α(β) = f α(β) α for all β C α. In addition, if otp(c α ) = λ, then for all j < λ: Im(f α C j α) = Im(f α ) = X α. Claim There exists some j < λ such that for every club D λ +, there exists some limit α < λ + with otp(c α ) = λ and C j α D. 5 Validity in a structure here has the obvious meaning, cf. [5, p. 891].

5 SQUARE WITH BUILT-IN DIAMOND-PLUS 5 Proof. Suppose not. Then for all j < λ, we may pick a club counterexample D j λ +. Let C = j<λ D j. By Clauses (5) and (6) of Definition 2.3, then, there must exist some α < λ + with otp(c α ) = λ and C α * C. Pick j < λ such that Cα j C. In particular, Cα j D j contradicting the choice of D j. Let j < λ be given by the preceding claim. For all α < λ +, let { Cα Cα, j if otp(c α ) > j; = C α, otherwise. Put fα = f α Cα, and Xα = X α. Then (Cα, Xα, fα) α < λ + forms a * λ-sequence with the additional desired property. Lemma 2.6. For every infinite cardinal λ, λ = + λ = * λ = λ. Proof. Let λ be an arbitrary infinite cardinal. The implication λ = + λ is trivial. To see that + λ = * λ, let (C α, N α, f α ) α < λ + be a + λ -sequence. For all α < λ+, let { P(α) N α, if otp(c α ) < λ; X α = Im(f α ), otherwise. Then (C α, X α, f α ) α < λ + forms a * λ -sequence. Out next task is showing that * λ = λ. By [3], (ω 1 ) implies ω, hence we hereafter assume that λ is uncountable. In particular, ω λ = λ. Let (C α, X α, f α ) α < λ + be given by Lemma 2.5. Fix a bijection ψ : λ λ + λ +. Let E = {α < λ + ψ[λ α] = α}. For every α < λ +, let π α : otp(c α ) C α denote the monotone enumeration of C α, and then for i < λ, write S i α = {τ < α ψ(i, τ) f α (π α (i))}. Claim There exists an i < λ such that for every club D λ + and every X λ +, there exists some limit α < λ + with otp(acc(c α )) = λ, such that S i α = X α and C α D. Proof. Suppose not, and so for all i < λ, pick a counterexample (D i, X i ). Let X = {ψ(i, τ) i < λ, τ X i }. By Clause (5) of Definition 2.2, we may find a club D E i<λ D i such that X α X α for all α D. Now, fix a limit ordinal α < λ + such that otp(c α ) = λ and C α D. In particular, α E. Note that since λ is uncountable, moreover otp(acc(c α )) = λ. Next, as otp(c α ) = λ, we infer from Clause (3) of Definition 2.2 the existence of some i < λ such that X α = f α (π α (i)). By definition of X and since π[λ α] = α, we conclude that S i α = {τ < α ψ(i, τ) X α} = X i α. By the choice of (D i, X i ), it must then be the case that C α D i. However, C α D D i. This is a contradiction. Let i < λ be given by the previous claim. For all α Eω λ+, let c α be a cofinal subset of α of order-type ω. Finally, for all α < λ +, let C α E, if sup(e C α ) = α; Cα = C α sup(e α), if sup(e α) < α; c α, otherwise,

6 6 ASSAF RINOT AND RALF SCHINDLER and S α = { Sα, i if sup(e C α ) = α;, otherwise. Claim (C α, S α) α < λ + is a λ -sequence. Proof. It is clear that for all limit α < λ +, C α is a club subset of α of order-type λ. Next, suppose that α < λ + and α acc(c α ). If sup(e C α ) = α, then C α = C α E and hence α acc(c α ), and sup(e C α ) = sup(e C α α) = α. Consequently, C α = C α E = C α α, f α (π α (i)) = f α (π α (i)) α, and S α α = S i α α = {τ < α ψ(i, τ) f α (π α (i))}. As sup(e C α ), we get in particular that α E, and hence the right hand side of the preceding is equal to {τ < α ψ(i, τ) f α (π α (i)) α} = S i α = S α. If sup(e α) < α, then C α = C α sup(e α), and α acc(c α ). Consequently, C α = C α sup(e α) = C α α sup(e α) = C α α sup(e α) = C α α, and S α α = = S α. Finally, let X λ + and a club D λ + be arbitrary. By the choice of i, let us pick a limit ordinal α < λ + with otp(acc(c α )) = λ such that Sα i = X α and C α D E. Then Sα = Sα i = X α, as sought. 3. The strongest principle holds in L Theorem 3.1. λ holds in L for all infinite cardinals λ. Proof. We follow the proof of [18, Theorem 11.64], making use of arguments from [1], [11], and [14]. We assume V = L. Let λ denote an arbitrary infinite cardinal. Consider the set C = {α < λ + J α Σω J λ +}, which is a club subset of λ + consisting of limit ordinals above λ. Note that C E>ω λ+ acc(c). Let φ : λ + C be the monotone enumeration of C. Let α C. Obviously, λ is the largest cardinal of J α. Note that as J α = ZFC, we have ρ ω (J α ) = α. We may therefore define ν(α) to be the largest ν > α such that α is a cardinal in the viewpoint of J ν. 6 Note also that if β < α, then by J α = J β = λ, we have that P(λ) J β P(λ) J α. So, ν(β) < α < ν(α) for all β C α. As ρ ω (J ν(α) ) = λ, let n(α) be the unique n < ω such that λ = ρ n+1 (J ν(α) ) < α ρ n (J ν(α) ). Let us write R for the set of all α < λ + such that ν(φ(α)) = ν + ω for some ν such that α is the only cardinal of J ν strictly above λ. For all α < λ +, set { J N α = ν(φ(α))+ω, if α R; J ν(φ(α)), otherwise. As ν φ is an increasing function from λ + to λ +, we have just established Clause (7) of Definition In particular, Jν+ω can see that α has cardinality λ.

7 SQUARE WITH BUILT-IN DIAMOND-PLUS 7 Let us now define a λ -sequence by what became the standard construction, cf. e.g. [18, pp. 270ff.], modulo various crucial adjustments. As we ll have to refer to some details of this construction later on in this proof, let us repeat this construction here for the convenience of the reader. First, for α C, we define D α as follows. We let D α consist of all α C α such that n( α) = n(α) and there is a weakly rσ n(α)+1 -elementary embedding σ : J ν( α) J ν(α) such that σ α = id, σ(p n( α)+1 (J ν( α) )) = p n(α)+1 (J ν(α) ), and if α J ν( α), then α J ν(α) and σ( α) = α. It is easy to see that if α D α, then there is exactly one map σ witnessing this, namely the one which is given by (1) h n( α)+1,p n( α)+1(j ν( α) ) J ν( α) (i, x) h n(α)+1,p n(α)+1(j ν(α) ) (i, x), where i < ω and x [λ] <ω. We here make use of the notation for fine structural iterated Σ 1 Skolem functions as presented e.g. in [18, Equation (11.29) on p. 252]. We shall denote the unique map as given by (1) by σ α,α. Notice that if α C, then so that if α D α, then J ν(α) J ν(α) = h n(α)+1,p n(α)+1(j ν(α) ) J ν(α) (ω [λ] <ω ), Im(σ α,α ) h n(α)+1,p n(α)+1(j ν(α) ) J ν(α) (ω [λ] <ω ), which means that there must be i < ω and x [λ] <ω such that the left hand side of (1) is undefined, whereas the right hand side of (1) is defined. Also notice that the maps σ α,α commute, i.e., if α D α and α D α, then α D α and Having constructed D α α C, we claim: σ α,α = σ α,α σ α,α. Claim Let α C. All of the following hold true: (a) D α is closed; (b) If cf(α) > ω, then D α is unbounded in α; (c) If α D α then D α α = D α. Proof. This is Claim of [18]. Notation. For α C, i < ω and x [λ] <ω, we shall denote: h α (i, x) = h n(α)+1,p n(α)+1(j ν(α) ) J ν(α) (i, x). Let α C. If sup(d α ) < α, let θ(α) = 0. Now, suppose sup(d α ) = α. We shall obtain some limit ordinal θ(α), and sequences μ α i i θ(α) and ξ α i i < θ(α), by recursion, as follows. Set μ α 0 = min(d α). Given μ α i with μ α i < α, we let ξ α i be the least ξ < λ such that h α (k, x) / Im(σ μ α i,α) for some k < ω and some x [ξ] <ω. Given ξ α i, we let μα i+1 be the least α D α such that h α (k, x) Im(σ α,α ) for all k < ω and x [ξ α i ]<ω such that h α (k, x) exists. Given μ α j j < i, where i is a limit ordinal, we set μα i = sup j<i μ α j.

8 8 ASSAF RINOT AND RALF SCHINDLER Naturally, θ(α) will be the least i such that μ α i For any α C, denote E α = {μ α i i < θ(α)}. = α. Claim Let α C. All of the following hold true: (a) ξi α i < θ(α) is a strictly increasing sequence of ordinals below λ; (b) otp(e α ) = θ(α) λ; (c) θ(α) > 0 iff E α is a club in α; (d) If α acc(e α ), then E α α = E α. Proof. (a) is immediate, and it implies (b). (c) is trivial. (d) Let α acc(d α ). We have E α D α, and D α = D α α by Claim 3.1.1(c). We now show that μ α i i < θ( α) = μ α i i < θ( α) and ξ α i i < θ( α) = ξ α i i < θ( α) by induction: Say μ α i = μ α i, where i + 1 θ( α) θ(α). Write μ = μ α i = μ α i. As σ μ,α = σ α,α σ μ, α, for all k < ω and x [λ] <ω, h α (k, x) Im(σ μ, α ) = h α (k, x) Im(σ μ,α ). This gives μ α i+1 μ α i+1 On the other hand, Im(σ μ α i+1,α ) contains the relevant witness so as to guarantee conversely that μ α i+1 μα i+1. Recall that φ : λ + C denotes the monotone enumeration of C. For all α acc(λ + ), we have φ(α) acc(c), so we set C α = φ 1 E φ(α). Since E φ(α) D φ(α) C = Im(φ), we have otp(c α) = θ(φ(α)) λ, and C α is a club in α iff θ(φ(α)) > 0. Because E μ D μ C μ for every μ C, we have that if α acc(c α), then C α = C α α. Claim Let α Eω λ+ be such that φ(α) = α. Then there exists a cofinal subset d of α of order-type ω satisfying the following: (a) If c J ν(α) is a club in α, then d * c; (b) d N α whenever α < α < λ +. Proof. The argument is a simplified version of the proof of [11, Theorem 4.15]. Fix α Eω λ+ such that φ(α) = α, so that α C. Recall that λ = ρ n(α)+1 (J ν(α) ) < α ρ n(α) (J ν(α) ). Let us write ν = ν(α), n = n(α), and τ = sup({ξ < ν J ν = ξ α}). I.e., either α is the largest cardinal of J ν in which case τ = ν, or else τ = α +Jν. Case 1. cf(τ) = ω. Let (τ m m < ω) be a sequence witnessing cf(τ) = ω. For each m < ω, we may inside J ν pick some club c m α such that c m * c for every club c α, c J τ m. E.g. we may let c m be the diagonal intersection of all clubs c α, c J τ m, as being given by some enumeration in J ν in order type α. Let then (ρ m m < ω) be a strictly increasing sequence which is cofinal in α and such that for every m < ω, ρ m k m cm. Set d = {ρ m m < ω}. Then for every c J ν which is a club in α, we have d * c. Case 2. cf(τ) > ω.

9 SQUARE WITH BUILT-IN DIAMOND-PLUS 9 By [14, Lemma 1.2], we must then have that ρ n (J ν ) = α, and if l < n is largest such that ρ l (J ν ) > α, then cf(ρ l (J ν )) = cf(τ) > ω. We then have l + 1 n and ρ l (J ν ) > ρ l+1 (J ν ) = ρ n (J ν ) = α > ρ n+1 (J ν ) = λ. Let (α m m < ω) be a sequence witnessing cf(α) = ω. By cf(τ) ω, there must be some m < ω such that h J l ν (α m {p(j l ν)}) τ is cofinal in τ. Note that as ω = cf(α) < cf(ρ l (J ν )) and α is a regular cardinal in J ν, cannot be cofinal in α. Set h J l ν (α m {p(j l ν)}) α H = h J l ν (α m {p(j l ν)}) and α = sup(h α). Let us write θ = cf(ρ l (J ν )) = cf(τ), and let (τ i : i < θ) be a strictly increasing sequence witnessing cf(τ) = θ, where τ i H for every i < θ. For each i < θ, in much the same way as in Case 1 we may inside H pick some club c i α such that c i * c for every c J τ i which is a club in α. Notice that we may arrange c i H by τ i H. We may assume without loss of generality that c i J τ i+1 for every i < θ. We then get H = c i * c j for all j < i < θ, which by the choice of α readily implies that c i α c j α for all j < i < θ. But as cf(α) θ, this gives that c = i<θ (c i α) is club in α. We may then let d be a cofinal subset of c of order type ω. Then for every club c J ν which is a club in α, we have d * c. We thus in both cases found a set d which satisfies (a) from Claim The existence of some such d is a Σ 1 statement in the parameter J ν(α). Therefore, if we pick d < L least with (a), then by N α J φ(α ) J λ + for α > α we will also get (b) from Claim Let us now for each α Eω λ+ define c α α as follows. If φ(α) = α, then we let c α be the < L least d which satisfies the conclusion of Claim If φ(α) α, then just let c α be the < L -least cofinal subset of α of order-type ω. Finally, for all α < λ +, let, if α = 0; {β}, if α = β + 1; C α = C α, if α acc(λ + ) & θ(φ(α)) > 0; c α, otherwise. Clearly, C α is a club in α of order-type λ. By Claim 3.1.1(c) and Claim 3.1.2(d), if α acc(c α ), then C α = C α α. Altogether, C α α < λ + satisfies Clauses (1) and (4a) of Definition 2.3. Let Γ: OR ω [OR] <ω be some simply-definable enumeration of ω [OR] <ω. Using Kleene s notation, for α < λ +, let g α be the partial function with dom(g α ) λ such that g α (χ) h φ(α) (Γ(χ)).

10 10 ASSAF RINOT AND RALF SCHINDLER Claim Let α < λ +. All of the the following hold: (a) J ν(φ(α)) = {g α (i) i < λ, g α (i) is defined}; (b) If sup(acc(c α )) = α, χ < otp(c α ), and g α (χ) is defined, then there exists some α acc(c α ) such that g α (χ) is defined; (c) If α acc(c α ), χ < otp(c α ) and g α (χ) φ( α) 2, then g α (χ) φ(α) 2 and g α (χ) = g α (χ) φ( α). Proof. (a) Since J ν(φ(α)) = h φ(α) (ω [λ] <ω ), as pointed out earlier. (b) is clear from the construction, as J ν(φ(α)) results from the direct limit of the system (J ν(φ(β)), σ φ(β),φ(β ) β β C α ). (c) Let α, α, and χ be as in the hypothesis of (c). Then g α (χ) φ(α) 2 follows immediately from the fact that g α (χ) φ( α) 2, as σ φ( α),φ(α) is Σ 0 elementary and sends its critical point φ( α) to φ(α). Moreover, if ξ < φ( α), then g α (ξ) = σ φ( α),φ(α) (g α )(ξ) = σ φ( α),φ(α) (g α (ξ)) = g α (ξ). We have already defined (C α, N α ) α < λ +, and our next goal is to define f α α < λ +. If λ = ℵ 0, then for every α acc(λ + ), we have acc(c α ) =, and hence we simply let f α : C α P(α) N α be the < L -least surjection. For α λ + acc(λ + ), we let f α : C α {0} be constant. If λ > ℵ 0, we do the following. Let ψ : λ λ be the < L -least function such that ψ 1 {i} is cofinal in λ for all i < λ. Given α acc(λ + ), let φ α : C α otp(c α ) denote the order-preserving isomorphism. Then, for all β C α, set and Z β α = {δ acc(c α ) β ψ(φ α (β)) < φ α (δ) & g δ (ψ(φ α (β))) φ(δ) 2}, f α (β) = { {ε < α g α (ψ(φ α (β)))(ε) = 1}, if Zα β ;, otherwise. For α λ + acc(λ + ), let f α : C α {0} be constant. Having constructed f α α < λ +, we claim: Claim Let α < λ + be arbitrary. All of the following hold: (a) f α is a (well-defined) function from C α to P(α) N α. Moreover, Im(f α ) J ν(φ(α)) ; (b) If α acc(c α ) and β C α, then f α (β) = f α (β) α; (c) If otp(c α ) = λ, then Im(f α ) = P(α) J ν(φ(α)) ; (d) f α is surjective iff α λ + R <, where R < = R {α < λ + otp(c α ) < λ}. Proof. To avoid trivialities, assume λ > ℵ 0 and α acc(λ + ). (a) Let β C α be arbitrary. If Z β α =, then f α (β) = which is indeed an element of P(α) J ν(φ(α)). Suppose that Z β α. Write χ = ψ(φ α (β)). Fix δ Z β α. Then δ acc(c α ) and χ < φ α (δ). Consequently, χ < otp(c δ ) and g δ (χ) φ(δ) 2, and then by Clause (c) of Claim 3.1.4, g α (χ) φ(α) 2. By Clause (a) of Claim 3.1.4, g α (χ) J ν(φ(α)). Since the latter is rud-closed, it follows that f α (β) P(α) J ν(φ(α)). By definition of N α (cf. page 6), we have J ν(φ(α)) N α. (b) Suppose α acc(c α ) and β C α. Then C α = C α α, φ α = φ α α, and ψ(φ α (β)) = ψ(φ α (β)), say, it is χ. As Z β α C α β and Z β α C α β, we altogether infer that Z β α = Z β α. In particular, if the latter is empty, then f α (β) = = f α (β) α. Next, suppose that Z β α is nonempty, and fix a witnessing element δ. By δ Z β α, we know that χ < otp(c δ ) and g δ (χ) φ(δ) 2, and then by Clause (c) of Claim 3.1.4, we know that g α (χ) φ( α) 2.

11 SQUARE WITH BUILT-IN DIAMOND-PLUS 11 By α acc(c α ) and χ < otp(c δ ) < otp(c α ) and Clause (c) of Claim 3.1.4, we then know that g α (χ) φ(α) 2 and g α (χ) α = g α (χ) α. Consequently, f α (β) = f α (β) α. (c) Suppose that otp(c α ) = λ, and let x be an arbitrary element of P(α) J ν(φ(α)). By x α φ(α), let ρ x : φ(α) 2 denote the characteristic function of x. Since φ(α) J ν(φ(α)) and since the latter is rud-closed, we have ρ x J ν(φ(α)). By Clause (a) of Claim 3.1.4, we may fix some ordinal i < λ such that g α (i) = ρ x. Since otp(c α ) = λ is an uncountable cardinal, we have sup(acc(c α )) = sup(c α ) = α, and then by Clauses (b) and (c) of Claim 3.1.4, let us fix a large enough α acc(c α ) such that i < otp(c α ) and g α (i) = g α (i) φ( α). Now, by the choice of ψ, there exists a large enough j with φ α ( α) < j < λ such that ψ(j) = i. As otp(c α ) = λ > j, let β C α be the unique element to satisfy φ α (β) = j. Then: Z β α = {δ acc(c α ) β i < φ α (δ) & g δ (i) φ(δ) 2}. By i < otp(c α ) = φ α ( α) and g α (i) φ( α) 2, we have α Z β α. So f α (β) = {ε < α g α (ψ(φ α (β)))(ε) = 1} = {ε < α g α (ψ(j))(ε) = 1} = {ε < α g α (i)(ε) = 1} = {ε < α ρ x (ε) = 1} = x, as sought. (d) If α R, then by Clause (a), Im(f α ) J ν(φ(α)) J ν(φ(α))+ω = N α. If otp(c α ) < λ, then Im(f α ) < λ = N α, so α is not onto. Finally, if α R and otp(c α ) = λ, then by Clause (c) and the definition of N α in this case, Im(f α ) = P(α) J ν(φ(α)) = P(α) N α. Thus, we have established Clauses (3) and (4b) of Definition 2.4. Claim Let α < λ +. All of the following hold: (a) {C β β < α} N α. Moreover: (b) {f β β < α} N α. Proof. Denote γ = φ(α) and ν = ν(γ). Clearly, α γ < ν and N α = J ν or N α = J ν+ω. Let β < α be arbitrary. (a) If C β = c β, then by γ C (β + 1), we get that J γ Σω J λ + and J γ = cf(β) = ω. In particular, if c β is the < L -least cofinal subset of β of order-type ω, then c β J γ J ν. If c β is not of this form, then it was obtained by Claim which also ensures that c β N α. Next, suppose that C β = C β. Notice first that (2) φ(β) < ν(φ(β)) < ν(φ(β)) + ω < φ(α) < ν(φ(α)) = N α OR. We have that C β = φ 1 E φ(β), where φ = φ β. By definition of C and since φ(β) C, we know that C φ(β) is Σ 1 -definable over J φ(β)+ω. By Equation (2), J φ(β)+ω N α and φ N α, and it suffices to show that E φ(β) N α. But an inspection of the construction of E φ(β) yields that E φ(β) is Σ 1 -definable over J ν(φ(β))+ω. Hence by Equation (2) again, E φ(β) N α. (b) We have already shown that C β N α. This immediately gives φ β N α. We have that g β is definable over J ν(φ(β)), so that g β N α by N β N α. Certainly, ψ N α. Taken together, f β N α. Having Clause (c) of Claim in mind, we now turn to verify Clause (5) of Definition 2.4. Claim Suppose that A λ + is some set, and B λ + is a club. Then there exists a club D {α C φ(α) = α} such that for all α D:

12 12 ASSAF RINOT AND RALF SCHINDLER (a) A α, B α J ν(α) ; (b) D α N α ; (c) C α * B. Proof. (a) Let h ω J denote the closure under Σ λ ++ n Skolem functions for J λ ++, for all n < ω, which are uniformly definable over all J γ, γ ω ω, in a canonical fashion. Let us recursively define X i i < λ + as follows. For i < λ +, let γ(i) be such that X 0 = h ω J λ ++ (ω [λ {A, B}]<ω ), X i+1 = h ω J λ ++ (ω [λ {A, B, X j j i }] <ω ), and X i = {X j j < i} for limit i > 0. π i : J γ(i) = Xi J λ ++, and write α(i) = X i λ + = λ +J γ(i). Notice that {C, φ} X 0, and hence α(i) is a closure point under φ, and φ 1 (α(i)) = α(i). Consider the set D = {α(i) i < λ + } which is a club subset of {α C φ(α) = α}. For i < λ +, J γ(i)+ω = α(i) is a cardinal, while the definition of ν(α(i)) entails that ν(α(i)) γ(i) + ω, and hence (3) ν(α(i)) > γ(i). Of course, A α(i) = π 1 i (A) J γ(i). Thus, A α J ν(α) for every α D, and likewise, B α J ν(α) for every α D. (4) (b) First, we point out that for all i < λ +, Equation (3) implies: π 1 i X j j < i N α(i). To see this, notice that the elementary embedding π i will respect the uniformly defined Σ n Skolem functions for J λ ++ and J γ(i), respectively, and hence π 1 π 1 i X 0 = h ω J γ(i) (ω [λ {A α(i), B α(i)}] <ω ), i X i+1 = h ω J γ(i) (ω [λ {A α(i), B α(i), πi 1 X j j i }] <ω ), and X i = {X j j < i} for limit i > 0. This gives that if α = α(i) D, then the sequence from (4) is 1 definable over J γ(i)+ω. However, we obviously have that in this situation α is the only cardinal of J γ(i), so that if γ(i) + ω = ν(α), then α R and N α OR = γ(i) + ω 2, hence Equation (4) holds true. If ν(α) > γ(i) + ω, then the sequence from Equation (4) is in J ν(α) N α. Thus, we have verified that D α N α for every α E. (c) Let α D be arbitrary, and we shall show that C α * B. By B α J ν(α), φ(α) = α and Claim 3.1.3, we may assume that C α c α. That is, C α = C α = φ 1 E α, and we must show that E α * φ B. Let i < λ + be such that α = α(i), and let μ E α be large enough such that (5) {γ(i), (φ B) α} Im(σ φ(μ),α ).

13 SQUARE WITH BUILT-IN DIAMOND-PLUS 13 Write γ = σ 1 φ(μ),α (γ(i)). We have that σ φ(μ),α J γ : J γ J γ(i) is fully elementary, so that in fact φ(μ) is a limit point of (φ B) α, and hence φ(μ) (φ B) α, i.e., μ B. As Equation (5) holds true for a tail end of μ E α, we have E α * φ B. We now verify Clause (6) of Definition 2.4, using an argument from [1, S2]. Assume (6) were to fail. Recalling Clause (d) of Claim 3.1.5, there are n, m < ω, some Π n m-sentence η valid in a structure (λ +,, A i i < ω ), and some club D λ + such that for every α D, α R < or else N α = η is not valid in (α,, A i α i < ω ). Let (D, A i i < ω ) be the < L least such pair. Notice that C, φ, and R < are all definable over J λ + +ω by some formulas with no parameters. Also, D and A i i < ω are both definable over J λ +n by some formulas with no parameters. But as λ + and λ +n, and hence J λ + +ω and J λ +n, are both Σ 1 definable over J λ +n+1 from the parameter λ +n, we get that D, A i i < ω, C, and φ, and also λ, λ +, λ +2,..., λ +n are all Σ 1 definable over J λ +n+1 from the parameter λ +n. Let us here and in what follows use the notation from [18, p. 194] which for X J γ writes h Jγ (X) for h Jγ (ω [X] <ω ), where h Jγ is the canonical Σ 1 Skolem function for J γ. We now have, setting D * = φ D, (6) Let ν be such that {D, A i i < ω, C, φ, D *, λ, λ +, λ +2,..., λ +n } h Jλ +n+1 ({λ+n }). σ : J ν = hjλ +n+1 (λ {λ+n }) Σ1 J λ +n+1, and write α = σ 1 (λ + ) = crit(σ) and β = σ 1 (λ +n ). Of course, J ν = h Jν (λ {β}), so that ρ 1 (J ν ) = λ and p 1 (J ν ) * {β}. 7 However, if we had p 1 (J ν ) < * {β}, then β h Jν (β), so that λ +n h Jλ +n+1 (λ+n ); but it easily follows from the Condensation Lemma that h Jλ +n+1 (λ+n ) J λ +n. Therefore, p 1 (J ν ) = {β}. Obviously, ν(α) = ν. By {D, φ, D * } h Jλ +n+1 ({λ+n }), we have that D α, φ (α α), D * α = σ 1 (D, φ, D * ) J ν, so that α D D *, and we also get that α is a closure point of φ, and α = φ(α). By α = φ(α), we have ν(φ(α)) = ν. As A i i < ω h Jλ +n+1 ({λ+n }), A i α = σ 1 (A i ) for every i < ω. By elementarity then, (7) N α = J ν = η is valid in (α,, A i α i < ω ). Claim (a) There is no ξ < λ such that h Jν (ξ {β}) α is cofinal in α, so that in particular {ξi α i < θ(α)} is cofinal in λ; (b) For every i < θ(α), ξi+1 α = ξα i ; (c) θ(α) = λ; (d) α R <. Proof. Write d = {(n, x) ω [λ] <ω h Jν (n, x {β}) is defined}. We now introduce the following notation. For ξ < λ, let us write ζ(ξ) λ +n+1 for the least ζ such that if (n, x) d (ω [ξ] <ω ), then h Jζ (n, ( x, λ +n )) is defined. As λ +n+1 is regular, ζ(ξ) < λ +n+1. (a) Assume that ξ < λ were such that h Jν (ξ {β}) α is cofinal in α. Using the map σ of Equation (6), h (ξ Jλ +n+1 {λ+n }) α is then cofinal in α. Let ζ = ζ(ξ) < λ +n+1. Trivially, d (ω [ξ] <ω ) h Jλ +n+1 (λ), so that ζ h Jλ +n+1 ({d (ω [ξ]<ω ), λ +n }) h Jλ +n+1 (λ {λ+n }) = Im(σ), 7 Here, * is the canonical well ordering of finite sets of ordinals, cf. [18, Problem 5.19 and p. 254].

14 14 ASSAF RINOT AND RALF SCHINDLER so that by using σ again, α = sup(h Jσ 1 (ζ) (ξ {β}) α). However, h Jσ 1 (ζ) (ξ {β}) J ν and α is regular in J ν. This is a contradiction. (b) This follows from the proof of (a). We have that and then ζ(ξ α i ) h Jλ +n+1 ({d (ω [ξα i ] <ω ), λ +n }), μ := sup(h Jν (ξ α i {β}) α) = sup(h Jσ 1 (ζ(ξ α i )) (ξα i {β}) α) h Jν ({d (ω [ξ α i ] <ω ), β}). However, by the Condensation Lemma, d (ω [ξ α i ]<ω ) h Jλ +n+1 ((ξα i )+ ), so that μ h Jν ((ξ α i ) + {β}). But this means that there is (n, x) ω [(ξ α i )+ ] <ω such that μ = h Jν (n, ( x, β)). But μ / h Jν (ξ α i {β}), which now readily gives ξ α i+1 < (ξα i )+. (c) By Clause (b), for every i < θ(α), otp({j < θ(α) ξ α j < ξ α i + }) = ξ α i +, which together with Clause (a) gives that θ(α) = λ. (d) As ν is certainly a limit of limit ordinals, α / R. But then Clause (c) gives α / R <. Altogether α D R <, contradicting Equation (7). 4. Applications 4.1. Preliminaries. For a family of functions T and a set of ordinals D, write T D = {f T dom(f) D}, and succ ω (D) = {δ D 0 < otp(d δ) < ω}. Definition 4.1. We say that T is a κ-tree, whenever there exists a set Ω of size κ, for which T <κ Ω; {dom(f) f T } = κ; for every f T, we have {f α α < κ} T ; T α := {f T dom(f) = α} has size < κ for all α < κ. A κ-aronszajn tree is a κ-tree with no cofinal branches. A κ-kurepa tree is a κ-tree admitting at least κ + many cofinal branches. A λ + -tree is special if it may be covered by λ many antichains. Following [2], which generalizes the case λ = ℵ 0 from [7], we say that a λ + -tree T is almost Souslin if for every antichain A T, the set {dom(z) z A} Ecf(λ) λ+ is nonstationary. Of course, almost Souslin and special are contradictory concepts. In [3],[4], the parameterized principle P (κ, μ, R, θ, S, ν, σ, E) was introduced and studied in relation with κ-souslin tree constructions. Here, we shall only give the definition of the special case which, for simplicity, is denoted by (S). (κ, μ, R, θ, S, ν, σ, E) = (κ, 2,, κ, {S}, 2, ω, (P(κ)) 2 ), Definition 4.2. For any regular uncountable cardinal κ, and stationary S κ, (S) asserts the existence of a sequence C α α < κ such that: C α is a club subset of α for every limit ordinal α < κ;

15 SQUARE WITH BUILT-IN DIAMOND-PLUS 15 C α = C α α for every ordinal α < κ and every α acc(c α ); for every sequence A i i < κ of cofinal subsets of κ, there exist stationarily many α S such that sup{β < α succ ω (C α β) A i } = α for all i < α. Note that for S S κ, every (S)-sequence is also a (S )-sequence. Clearly, every (κ)- sequence is in particular a (κ)-sequence. Definition 4.3. The principle λ (S) asserts the existence of a (S)-sequence C α α < sup(s) with the additional property that otp(c α ) λ for all α. Note that every λ (λ + )-sequence is in particular a λ -sequence. Definition 4.4 ([2]). Suppose that T <κ Ω is a κ-tree, and S is stationary in κ. We say that T is (S)-respecting if there exist a subset S S and a sequence b α : T C α α Ω { } α < κ such that: (1) C α α < κ is a (S)-sequence; (2) T α Im(b α ) for every α S; (3) if α acc(c α ) and x T C α, then b α (x) = b α (x) α. The notion of λ (S)-respecting is defined in a similar fashion Walks on ordinals. In this subsection, we address the trees obtained from walks on ordinals, as introduced in [21] (see also [23]). Suppose that C = C α α < κ is a C-sequence over some fixed regular uncountable cardinal κ. That is, C α is a club in α for all limit α < κ, and C α+1 = {α} for all α < κ. Recall few of the characteristic functions of walks on ordinals: Definition 4.5 ([21],[23]). Define Tr : [κ] 2 ω κ, ρ 2 : [κ] 2 ω, ρ 1 : [κ] 2 κ and ρ 0 : [κ] 2 <ω κ as follows. For all α < δ < κ, let δ, n = 0; Tr(α, δ)(n) := min(c Tr(α,δ)(n 1) α), n > 0, & Tr(α, δ)(n 1) > α; α, otherwise; ρ 2 (α, δ) := min{n < ω Tr(α, δ)(n) = α}; ρ 1 (α, δ) := max(ρ 0 (α, δ)), where ρ 0 (α, δ) := otp(c Tr(α,δ)(i) α) i < ρ 2 (α, δ). Definition 4.6 ([15]). Define φ 2 : [κ] 2 2 by stipulating φ 2 (α, δ) = 1 iff α acc(c Tr(α,δ)(ρ2 (α,δ) 1)). Definition 4.7 ([21],[23]). For all δ < κ, let ρ 0δ : δ <ω δ, ρ 1δ : δ δ and ρ 2δ : δ ω denote the fiber maps α ρ 0 (α, δ), α ρ 1 (α, δ) and α ρ 2 (α, δ), respectively. Then, put T (ρ 0 ) := {ρ 0δ β β δ < κ}; T (ρ 1 ) := {ρ 1δ β β δ < κ}; T (ρ 2 ) := {ρ 2δ β β δ < κ}. It is easy to see that if {C α β α < κ} < κ for all β < κ, then T (ρ 0 ), T (ρ 1 ) and T (ρ 2 ) are κ-trees. Fact 4.8 ([21],[23]). Suppose that T (ρ 0 ) is derived from walks along a λ -sequence, 8 then T (ρ 0 ) is a special λ + -Aronszajn tree. 8 This means that the C-sequence that was used to define ρ0 in Definition 4.5 is a λ -sequence, C α α < λ +.

16 16 ASSAF RINOT AND RALF SCHINDLER Fact 4.9 ([21],[23]). Suppose that λ is a regular cardinal, and T (ρ 1 ) is derived from walks along a C-sequence C α α < λ + for which otp(c α ) λ for all α < λ +. Then: (1) T (ρ 1 ) <λ+ λ; (2) for every z T (ρ 1 ) and i < λ, the set z 1 {i} has size < λ; (3) for every γ < δ < λ +, the set {ξ γ ρ 1γ (ξ) ρ 1δ (ξ)} has size < λ. Theorem If C is a (S)-sequence, then the corresponding trees T (ρ0 ), T (ρ 1 ), T (ρ 2 ) are (S)-respecting, as witnessed by the very same C. Proof. Suppose that C = C α α < κ is a (S)-sequence. Fix bijections π : κ 2 κ and ψ : κ <ω {C α β α, β < κ}. By the coherence of C, we have {C α β α < κ} < κ for all β < κ, and hence, the set E := {β < κ π[β] = 2 β & <ω {C α β α < κ, sup(c α β) < β} = ψ[β]} is a club in κ. Fix a surjection φ : κ κ such that the preimage of any singleton is cofinal in κ. Put S := {α S E α φ[c α ]}. Claim C is a (S)-sequence. Proof. Given a sequence A i i < κ of cofinal subsets of κ, write: { A A j, if π(i) = (0, j); i := φ 1 {j}, if π(i) = (1, j). As C is a (S)-sequence, the following set is stationary: R := {α S E i < α sup{β C α succ ω (C α β) A i} = α}. Let α R be arbitrary. By π[α] {0} α, we have sup{β C α succ ω (C α β) A i } = α for all i < α. By π[α] {1} α, we have sup{β C α succ ω (C α β) φ 1 {i}} = α for all i < α. In particular, α S. For all ε < κ and nonzero limit β < κ, let ψε β denote ψ(ε) C β. Then define Σ β ε : β <ω κ, as follows. Given α < β, put j = min{j dom(ψ β ε ) ψ β ε (j) α }; α + = min(ψ β ε (j ) α), and Σ β ε (α) = otp(ψ β ε (j) α) j j otp(c Tr(α,α + )(i+1) α) i + 1 < ρ 2 (α, α + ). Let m : <ω κ κ denote a map that satisfies σ max(σ) for all nonempty sequence σ. Let l : <ω κ ω denote the map that satisfies σ σ for all sequence σ. Denote Ω 0 := <ω κ, Ω 1 := κ and Ω 2 := ω. For all i < 3, define b i = b β i : T (ρ i ) C β β Ω i β < κ by stipulating: b β 0 (x) = Σβ φ(dom(x)), b β 1 (x) = m Σβ φ(dom(x)), b β 2 (x) = l Σβ φ(dom(x)). Claim Suppose i < 3, β acc(c β ) and x T (ρ i ) C β. Then b β i (x) = bβ i (x) β.

17 SQUARE WITH BUILT-IN DIAMOND-PLUS 17 Proof. By β acc(c β ), we have C β = C β β, and it suffices to show that Σ β ε = Σ β ε β for all ε < κ. But the latter is straight-forward to verify. Claim T (ρ i ) β Im(b β i ) for every i < 3 and β S. Proof. We concentrate on the case i = 0. Let β S and z T (ρ 0 ) β be arbitrary. Pick δ [β, κ) such that z = ρ 0δ β. Let n = ρ 2 (β, δ) φ 2 (β, δ). Define σ : n P(β), by stipulating σ(j) := C Tr(β,δ)(j) β. By β S E and the definition of n, there exists some ε < β such that ψ(ε) = σ. By β S, there exists some γ C β such that φ(γ) = ε. Let x = z γ. By z T (ρ 0 ) β, we have x T (ρ 0 ) γ, let alone x T (ρ 0 ) C β. We have φ(dom(x)) = ε, and so, to show that b β 0 (x) = z, it suffices to prove that Σβ ε = z. First, we make the following observation. If φ 2 (β, δ) = 0, then ψε β = ψ(ε) C β = σ C β = C Tr(β,δ)(j) β j < ρ 2 (β, δ) C β = C Tr(β,δ)(j) β j n. If φ 2 (β, δ) = 1, then β acc(c Tr(β,δ)(ρ2 (β,δ) 1)), and hence C β Consequently = C Tr(β,δ)(ρ2 (β,δ) 1) β. ψε β = ψ(ε) C β = σ C β = C Tr(β,δ)(j) β j < ρ 2 (β, δ) 1 C β = C Tr(β,δ)(j) β j < ρ 2 (β, δ) 1 (C Tr(β,δ)(ρ2 (β,δ) 1) β) = C Tr(β,δ)(j) β j n. Now, let α < β be arbitrary. By z = ρ 0δ β and definition of ρ 0δ, we have: z(α) = otp(c Tr(α,δ)(j) α) j < ρ 2 (α, δ). Let j = min{j dom(ψ β ε ) ψ β ε (j) α }. Then, for all j < j, we have (C Tr(β,δ)(j) β) α =, and hence min(c Tr(β,δ)(j) α) = min(c Tr(β,δ)(j) β). As Tr(α, δ)(0) = δ = Tr(β, δ)(0), it follows that Tr(α, δ) j + 1 = Tr(β, δ) j + 1, and hence Let α + := Tr(α, δ)(j ). By definition z(α) j + 1 = Σ β ε j + 1. z(α) = (z(α) j + 1) otp(c Tr(α,α + )(i+1) α) i + 1 < ρ 2 (α, α + ). By Tr(β, δ)(j ) = Tr(α, δ)(j ) = α +, we have min(ψ β ε (j ) α) = α +. Altogether: z(α) = Σ β ε (α). So for each i < 3, S and b i witness that T (ρ i ) is (S)-respecting. Theorem Suppose that * λ holds for a given regular uncountable cardinal λ. Then there exists a λ (Eλ λ+ )-sequence D α α < λ + satisfying the following. For every stationary S Eλ λ+, there are δ S and γ nacc(d δ ) S such that D δ γ D γ.

18 18 ASSAF RINOT AND RALF SCHINDLER Proof. Let (C α, X α, f α ) α < λ + witness * λ. Fix a surjection ψ : λ λ with the property that for all j < λ, the set {i < λ ψ (i, i + ω) = {j}} has size λ. We may assume that C α+1 = for all α < κ. Denote Λ = {α < λ + otp(c α ) = λ}. Clearly, Λ = Eλ λ+. Write κ = λ+. For all α < κ, let π α : otp(c α ) C α denote the monotone enumeration of C α. We now define a sequence of functions σ = σ α : otp(c α ) α α < κ by recursion over α < κ. For this, suppose α < κ, and σ α has already been defined. The definition of σ α : otp(c α ) α is obtained by recursion over i < otp(c α ). For this, suppose i < otp(c α ) and σ α i has already been defined. Let Xα i = {ξ f α (π α (ψ(i))) π α (i) < ξ π α (i + 1), ψ(i) i}, and Yα i = {ξ Xα i σ α (i + 1) = σ ξ (i + 1)}. Then, let min(yα i ), if i = i + 1, Yα i ; σ α (i) = min(xα i ), if i = i + 1, Yα i =, Xα i ; π α (i), otherwise. Put D α = Im(σ α ). Claim D α is a club in α, otp(d α ) = otp(c α ), acc(d α ) = acc(c α ) and if α acc(d α ), then D α = D α α. Proof. For all i < otp(c α ), we have (a) π α (i) < σ α (i + 1) π α (i + 1); (b) σ α (i) = π α (i) for all limit i < otp(c α ), including i = 0. So otp(d α γ) otp(c α γ)+1 for all γ < α, and acc(d α ) = acc(c α ). Towards a contradiction, suppose that α acc(d α ), while D α α D α. Let i < otp(c α ) be the least such that σ α (i) σ α (i). By α acc(d α ) = acc(c α ), we have C α = C α α, π α π α. So by Clause (b), i must be a successor ordinals, say i = i + 1. We have: X i α = {ξ f α (π α (ψ(i ))) π α (i ) < ξ π α (i + 1), ψ(i ) i } = {ξ f α (π α (ψ(i ))) α π α (i ) < ξ π α (i + 1), ψ(i ) i} = X i α. By minimality of i, we then also have Y i α = Y i α. But then σ α (i) = σ α (i). This is a contradiction. So D := D α α < κ is a λ -sequence. Next, we prove that D is a λ (S)-sequence for every stationary S Λ. Claim For every stationary S Λ, and every sequence A δ δ < κ of cofinal subsets of κ, there exist stationarily many α S such that for every δ < α, we have sup{β D α succ ω (D α β) A δ } = α. Proof. Let S and A δ δ < κ be as in the hypothesis. By Clause (5) of Definition 2.2, for every δ < κ, fix a club E δ acc + (A δ ) such that A δ α N α for all α E δ. Let C := δ<κ E δ. By Clause (5) of Definition 2.2, let us fix α S such that C α * C. Let ε, δ < α be arbitrary. We shall find β D α ε such that succ ω (D α β) A δ. Without loss of generality, ε is also large enough so that C α ε C. By δ α C, we have α E δ, and hence A δ α X α. Since α S Λ, we appeal to Clause (3) of Definition 2.2 to obtain some j < otp(c α ) such that f α (π α (j)) = A δ α. Fix a large enough i < otp(c α ) such that ψ (i, i + ω) = {j}, and π α (i) > max{ε, δ, π α (j)}. Write β := min(d α π α (i) + 1). Then β > ε and succ ω (D α β) = σ α (i, i + ω).

19 SQUARE WITH BUILT-IN DIAMOND-PLUS 19 Let i [i, i + ω) be arbitrary. We shall show that σ α (i + 1) A δ. We have X i α = {ξ f α (π α (ψ(i ))) π α (i ) < ξ π α (i + 1), ψ(i ) i } = = {ξ f α (π α (j)) π α (i ) < ξ π α (i + 1)} = = {ξ A δ α π α (i ) < ξ π α (i + 1)}. By π α (i +1) > max{ε, δ}, we have π α (i +1) E δ acc + (A δ ), and hence Xα i. Consequently, σ α (i + 1) Xα i A δ, as sought. Claim For every stationary S Λ, there are δ S and γ nacc(d δ ) S such that D δ γ D γ. Proof. Fix a surjection φ : κ {D α β α, β < κ}. Note that by Claim , the following set is a club D := {δ < κ {D α β α, β < κ, sup(d α β) < δ} = φ[δ]}. Let S be an arbitrary stationary subset of Λ. Define a function g : κ κ κ by letting for every α < κ: { sup{ξ S φ(α) D ξ } + 1, if sup{ξ S φ(α) D ξ } < κ; g(α, ζ) := min{ξ S ξ > ζ, φ(α) D ξ }, otherwise. Let E := {δ < κ g[δ δ] δ}. Now, fix δ S E and ε < δ satisfying the following: C δ ε D E; S δ X δ. Since δ Λ and S δ X δ, let us fix some j < otp(c δ ) such that f δ (π δ (j)) = S δ. Fix a large enough i < otp(c δ ) such that ψ(i) = j, and π δ (i) > max{ε, π α (j)}. By π δ (i + 1) D, let us fix some α < π δ (i + 1) such that φ(α) = D δ π δ (i + 1). By α δ E, we have g[{α} δ] δ. As φ(α) D δ and δ S, we thus infer that g(α, ζ) > ζ for all ζ < κ. In particular, by π δ (i + 1) E, we have π δ (i) < g(α, π δ (i)) < π δ (i + 1). Recalling that and X i δ = {ξ S δ π δ(i) < ξ π δ (i + 1)}, Y i δ = {ξ Xi δ σ δ (i + 1) = σ ξ (i + 1)} = {ξ X i δ φ(α) D ξ}, we see that g(α, π δ (i)) witnesses that Yδ i is nonempty. Write γ := σ δ(i + 1). Then γ Yδ i nacc(d δ ) S, and hence D δ γ D γ. We now address the trees T (ρ 0 ) and T (ρ 1 ). Note that the latter is a projection of the former. 9 Corollary Suppose that * λ holds for a given uncountable cardinal λ = λ<λ. Then there exists a λ (Eλ λ+ )-sequence which is respected by the corresponding trees T (ρ 0) and T (ρ 1 ). Moreover: (1) T (ρ 0 ) is a special λ + -Aronszajn tree; (2) T (ρ 1 ) is an almost Souslin, λ + -Aronszajn tree; (3) T (ρ 1 ) can be made special by means of a cofinality-preserving forcing. 9 Indeed, let m be some map satisfying σ max(σ) for every nonempty sequence of ordinals, σ. Then T (ρ1) is the image of T (ρ 0) under the map t m t.

20 20 ASSAF RINOT AND RALF SCHINDLER Proof. Let D = D α α < λ + be given by Theorem In particular, D is a λ (Eλ λ+ )-sequence. Let T (ρ 0 ) and T (ρ 1 ) denote the trees derived from walks along D. By Theorem 4.10, T (ρ 0 ) and T (ρ 1 ) are λ (Eλ λ+ )-respecting as witnessed by our D. (1) Since D is in particular a λ -sequence, we get from Fact 4.8 that T (ρ 0 ) is special. (2) It is easy to see that T (ρ 1 ) is a λ + -tree. By Clause (2) of Fact 4.9, T (ρ 1 ) is moreover Aronszajn. To see that T (ρ 1 ) is almost Souslin, suppose that A T (ρ 1 ), and that S = {dom(z) z A} Eλ λ+ is stationary. For all δ S, fix z δ A such that dom(z δ ) = δ. We now run the arguments from [10]. For all δ S, put y δ = z δ {ξ < δ z δ (ξ) ρ 1δ (ξ)}. By λ <λ = λ and Fact 4.9, we may find a stationary subset S S such that {y δ δ S } is a singleton. It follows that {z δ δ S } is an antichain iff {ρ 1δ δ S } is an antichain. So, let us show that the latter is not an antichain. By the choice of D, let us fix δ S and γ nacc(d δ ) S such that D δ γ D γ. Let α < γ be arbitrary, and we shall show that ρ 1γ (α) = ρ 1δ (α). Write β = sup(d δ γ). Clearly, β D γ D δ and D γ β = D δ β. If α < β, then there exists some β (D γ D δ (β + 1)) for which ρ 0δ (α) = max{otp(d δ α), ρ 0δ (β )} and ρ 0γ (α) = max{otp(d γ α), ρ 0γ (β )}. Since otp(d δ α) = otp(d δ β α) = otp(d γ β α) = otp(d γ α), we infer that ρ 1γ (α) = ρ 1δ (α). If β α < γ, then min(c δ α) = γ. Consequently ρ 1δ (α) = max{otp(c δ α), ρ 1γ (α)}. By definition, we have ρ 1γ (α) otp(c γ α). As C γ C δ γ, we have otp(c γ α) otp(c δ γ α) = otp(c δ α), and hence ρ 1δ (α) = ρ 1γ (α). (3) Let P denote the collection of all partial specializing functions of size < λ. That is, p P iff it is a function with dom(p) [T (ρ 1 )] <λ, Im(p) λ, such that p(y) p(z) for all y z in dom(p). Clearly, P is (< λ)-closed. It remains to verify that P has the λ + -cc. Towards a contradiction, suppose that P admits an antichain of size λ +. Then by λ <λ = λ and a standard -system argument, one could find some cardinal μ < λ and a family F [T (ρ 1 )] μ consisting of λ + many pairwise disjoint sets with the property that for every two distinct a, b F, there exist x a and y b such that x and y are comparable. For all a F, let {a(i) i < μ} be some enumeration of a. For every δ Eλ λ+, pick a δ F such that min{dom(x) x a} > δ, and define f δ : μ δ λ by stipulating f δ (i) = a(i) δ. Then, for all γ, δ Eλ λ+ there exist i, j < μ such that f γ (i) and f δ (j) are compatible. For all δ Eλ λ+, let D δ := {ξ < δ i < μ[f δ (i)(ξ) ρ 1δ (ξ)]}. By Clause (3) of Fact 4.9, D δ < λ. By λ <λ = λ, we may find a stationary set S Eλ λ+ for which {( f δ (i) D δ ) i < μ, (ρ 1δ D δ )) δ S} is a singleton. Consequently, {ρ 1δ δ S} forms a chain in T (ρ 1 ), contradicting the fact that T (ρ 1 ) is Aronszajn. If one is willing to give away the respecting feature of the preceding, then it is possible to relax * λ to just * (λ + ): Corollary Suppose that * (λ + ) holds for a given infinite cardinal λ = λ <λ. Then there exists a C-sequence for which the corresponding trees T (ρ 0 ) and T (ρ 1 ) satisfy Clauses (1) (3) of Corollary Notice that the same ideas of this section provides a proof to the following, which unlike Corollary 4.12, also apply to the case of λ singular: Corollary If λ holds for a given uncountable cardinal λ, then there exists a λ (E λ+ cf(λ) )- sequence which is respected by the corresponding trees T (ρ 0 ) and T (ρ 1 ). Moreover: