aa r X i v : . [ m a t h . L O ] M a r A Decomposition Theorem for Aronsza jn Lines
Keegan Dasilva Barbosa
Abstract
We show that under the proper forcing axiom the class of all Aron-szajn lines behave like σ -scattered orders under the embeddability re-lation. In particular, we are able to show that the class of better quasiorder labeled fragmented Aronszajn lines is itself a better quasi order.Moreover, we show that every better quasi order labeled Aronszajnline can be expressed as a finite sum of labeled types which are alge-braically indecomposable. By encoding lines with finite labeled trees,we are also able to deduce a decomposition result, that for every Aron-szajn line L there is integer n such that for any finite colouring of L there is subset L ′ of L isomorphic to L which uses no more than ncolours. It was shown by Carlos Martinez-Ranero that under PFA, the class of Aron-szajn lines is a better quasi order under the embeddability relation [2]. Theproof requires the development of an analogue to Hausdorff rank for scat-tered linear orders [4] to the Aronszajn case, as well as the construction ofincompatible Aronszajn lines D + α and D − α , α ∈ ω , that behave as universallines of rank ≤ α . What is interesting about the lines D + α and D − α is thatthey are fairly homogeneous. In particular, they are algebraically indecom-posable and are recursively constructed via a variant of shuffle described byLaver [6]. Laver used shuffles to recursively construct the class of σ -scatteredlinear orders and showed the class was BQO. We follow his construction tofind analogues of his results in the context of fragmented Aronszajn lines.In particular, we are able to recursively construct a well behaved class ofalgebraically indecomposable fragmented Aronszajn lines that behave as thebuilding blocks for all fragmented Aronszajn types.1 heorem 1.1. (PFA) Let Q be a BQO. Any Q labeled fragmented Aronszajnline Φ ∈ C ( Q ) can be written as a finite sum of members from a class of Q labeled algebraically indecomposable Aronszajn lines H ( Q ) . By showing that our constructed class H ( Q ) is BQO, we are able to givean alternative proof that the class of Aronszajn lines is BQO under PFA.More interestingly, we are also able to code algebraically indecomposableAronszajn lines onto BQO labeled trees like in [7] to show the followingdecomposition theorem for Aronszajn lines. Theorem 1.2. (PFA) For any Aronszajn type φ , there is an n ∈ ω such that ∀ k ∈ ω φ → ( φ ) k,n The paper is organized as follows. Section 2 will be devoted entirelyto preliminaries. If one is familiar with linear orders, BQO’s and Ramseydegrees, they can skip to section 2.4. While most of 2.4 is stating wellknown facts about Aronszajn lines, we also define shuffles here which willbecome highly relevant later. Section 3 is split into three major components.Section 3.1 is devoted to the finer structure analysis i.e defining shufflesand recursively constructing every Aronszajn line via shuffles of { , } . Wealso prove some relevant properties this class has. In the second, we proveTheorem 1.1 using Laver’s techniques from [6]. In the last, we prove theorem1.2 by adapting Laver’s finite tree coding argument [7]. We will first define what it means for a quasi order to be better. We will notuse the classical definition developed by Nash-Williams [11], but rather thetopological one seen in [1] and originally developed by Simpson [14]. Thisdefinition lends itself well to applications of the Galvin-Prikry theorem asseen in [2].
Definition 2.1.
Let Q be a quasi order. We say Q is a BQO if for any f : [ N ] ∞ → Q , Borel with respect to the discrete topology of Q , there existsan infinite A ⊆ N such that ∀ a ∈ [ A ] ∞ f ( a ) ≦ f ( a \ min a ).A nice property of BQO’s is they behave a lot like well orders. In partic-ular, we can do induction on them. 2 act 2.1. (BQO induction) Let Q be a BQO. To show a statement is truefor all q ∈ Q , we may suppose it is true for the set { r ∈ Q : q (cid:20) r } We will prove our desired results by coding our BQO’s onto trees andutilizing the infinite tree theorem as Laver did.
Definition 2.2.
We denote
F T Q T Q to be the class of all rooted trees thatare finite or have height ≤ ω respectively indexed by a quasi order ( Q, ≦ ).We define the following quasi orders ≦ , ≦ I ≦ s and ≦ m on T Q . • ( T, l ) ≦ ( S, m ) ⇐⇒ there exists a injective f : T → S such that f ( t ∧ t ) = f ( t ) ∧ f ( t ) and l ( t ) ≦ m ( f ( t )). • ( T, l ) ≦ I ( S, m ) ⇐⇒ T = S and l ( t ) ≦ m ( t ). • ( T, l ) ≦ s ( S, m ) ⇐⇒ there exists a map f : T → S such that t < T t ⇒ f ( t ) < S f ( t ) such that l ( t ) ≦ m ( f ( t )). • ( T, l ) ≦ m ( S, m ) ⇐⇒ there exists a map f : T → S such that t ≤ T t ⇒ f ( t ) ≤ S f ( t ) such that l ( t ) ≦ m ( f ( t )). Fact 2.2. (Infinite tree theorem) Q BQO ⇒ F T Q is BQO under ≦ and ≦ m , T Q is BQO under ≦ and ≦ s . [11] [6]Elements in these sets will be denoted as pairs ( T, l ), where T is a rootedtree and l : T → Q is the labeling. Definition 2.3.
Given a q ∈ Q , we define 1 q to be the one point tree indexedby q . Definition 2.4.
Given Q labeled trees T i indexed by some set i ∈ I , wedefine the tree T = [ q : T i i ∈ I ] to be the tree whose root is labeled by q andbranches into T i for each i ∈ I i.e the immediate successors of the root, arethe roots of T i . In the case I is linearly ordered, the lexicographical order of T will be determined by I in the natural way.It may not be clear why the ≦ m relation would be useful, as ≦ is alot simpler and easier to understand. However, ≦ m will allow us to useLaver’s covering theorem to take control of the number of treetops (sometimesreferred to as leaves) our finite trees will have for the proof of theorem 1.2.3 efinition 2.5. Let Q be a better quasi order. Take W ⊆ F T Q . We call W ′ ⊆ F T Q a cover of W if for every ( T, l ) ∈ W ∃ ( S, m ) ∈ W ′ such that S is a subtree of T and ( T, l ) ≡ m ( S, m ). We say W is n -coverable if there is acover W ′ for which each ( S, m ) ∈ W ′ has at most n treetops. Fact 2.3. (Laver’s covering theorem) Let Q be a better quasi order. Forevery W ⊆ F T Q , there is an n for which W is n -coverable. [7] First, we will start with some notation. The letters L and M will be reservedfor linear orders.
Definition 2.6.
Given a linear order L , we define its order type tp( L ) to bethe class of all linear orders isomorphic to L .Greek letters such as φ ψ and ϕ will be reserved for types. We willsometimes conflate linear orders with their types. We will try to do this asinfrequently as possible, though we will ignore this rule entirely for specialtypes like the rational type Q , regular cardinals κ and the minimal Country-man types C , C ∗ . We now define the main quasi order on types that will beof interest to us. Definition 2.7.
Given two linear orders L and M of type φ and ψ respec-tively, we say φ ≦ ψ if there exists an embedding of L into M . If φ ≦ ψ and ψ ≦ φ , we say φ ≡ ψ .The equivalence relation ≡ is some times referred to as the biembeddabil-ity relation. There are many algebraic operations one can define on types.In particular, we can define sum and product Definition 2.8.
Given L and M of type φ and ψ respectively, we define φ · ψ as the order type of L × M with the antilexicographical ordering, Definition 2.9.
Given L and M of type φ and ψ respectively, we define φ + ψ to be the order type of L × { } ∪ M × { } with the antilexicographicalordering.Interestingly, there is a way to describe linear ordered sums of types.This, for one gives us a lot more tools to algebraically analyze and constructtypes. Secondly, it also generalizes both finite sum and product.4 efinition 2.10. Given a linear order L and a collection of types { φ x : x ∈ L } , we define the type P x ∈ L φ x to be tp( S x ∈ L M x × { x } ), where tp( M x ) = φ x and S x ∈ L M x × { x } is ordered antilexicographically.One can check that P x ∈ L φ ≡ φ · tp( L ). Moreover, finite sums φ + .. + φ k can be seen as the ordered sum φ i over the linear order { , ..., k } with thenatural order. We also have a dual/reverse operation ∗ Definition 2.11.
Given a type φ , the reverse φ ∗ is the type of ( L, > ), wheretp(
L, < ) = φ .Often to analyze an L -sum of types, it will be beneficial to break L apartinto disjoint convex pieces and work with the natural order they inherit from L . We will refer to this order as the block order. Definition 2.12.
Given a linear order (
L, < ) and a collection of disjointintervals I , the block order on I is defined by A < B ⇐⇒ ∀ x ∈ A ∀ y ∈ Bx < y .We will also need to work with labeled orders. They are defined nearidentically to labeled trees.
Definition 2.13.
Given a quasi order ( Q, ≦ ) and a order type φ , we callΦ = ( L, l ) a Q labeled φ type if tp( L ) = φ and l : L → Q . Given two labeledtypes Φ = ( L, l ) and Ψ = (
M, m ), we say Φ ≦ Ψ if there exists an embedding f : L → M such that l ( x ) ≦ m ( f ( x )). For q ∈ Q , 1 q denotes the one pointedorder labeled by q . Our interest in the final portion of this paper will be on decomposition prop-erties held by Aronszajn lines. In particular, we will be interested in provingthe existence of a Ramsey degree.
Definition 2.14.
Given a linear order L , we write L → ( L ) k,n to mean thatfor k ∈ N and any colouring c : L → k , there exists A ∈ [ k ] n such that c − ( A )contains an isomorphic copy of L . If this holds for all k , we say L has bigRamsey degree bounded by n . 5 act 2.4. If L → ( L ) k,n and tp( L ) ≡ tp( M ), then M → ( M ) k,n .Consequently, rather than speak about a particular linear order, we caninstead adopt the notation ∀ k ∈ ω φ → ( φ ) k,n to mean any linear order oftype φ has big Ramsey degree bounded by n. Lemma 2.1.
Given order types φ, ψ , if φ → ( φ ) k,n and ψ → ( ψ ) k,n , then φ + ψ → ( φ + ψ ) k,n + n . Many well known orders have Ramsey degree bounded by 1. For example,the rationals Q , every regular cardinal and the generalized rationals η αβ [6]. Aweaker type of Ramsey degree is the property of algebraic indecomposability. Definition 2.15.
We call a type φ algebraically indecomposable (AI forshort) if whenever φ ≡ ψ + ψ , there is an i ∈ { , } for which φ ≦ ψ i In particular, being AI means having Ramsey degree bounded by 1 whenwe restrict our class of colourings to convex ones. For the class of σ -scatteredorders, the subclass of AI types behave as the building blocks. We will showlater that this remains true for Aronszajn types under PFA. An Aronszajn line is any line of size ℵ that is not isomorphic to a subor-der of the reals R and does not embed ω or ω ∗ . There have been manyconstructions of Aronszajn lines. A special subclass of Aronszajn lines areCountryman lines. We call a lines C Countryman if its square C × C (viewedas a product of posets, not linear orders) can be decomposed into countablymany chains. Two classic examples of Countryman lines can be found in [13]and [15]. The latter of the two is also minimal in that for any Aronszajn line A contains a copy of it or its reverse as a suborder. We will fix the name C for this line. Fact 2.5.
For any L ∈ { C, C ∗ , Q } , there exists a collection of disjoint inter-vals I which is isomorphic to L under the block order.The above fact will be vital for us. It is trivial to prove for Q , whilefor C it requires a small forcing argument. In particular, it suffices to showthat C (and consequently C ∗ ) contain no Souslin suborder. One can find theargument in [2] along with the following.6 act 2.6. (MA ℵ ) For any L ∈ { C, C ∗ , Q } , L ≡ L .Note, this is generally true for Q and does not rely on Martins axiom.Another critical fact we will need is the existence of a universal Aronszajnline. Having one will mean that we can construct an ω sequence of AIAronszajn lines cofinal under ≦ that are constructible from C under ouralgebraic operations. More on this will appear in the next section. Fact 2.7. (PFA) Every Aronszajn line is either universal, or fragmented. [9]In order to use these two facts, for the rest of the paper, we shall beassuming PFA.
We first start by defining a shuffle.
Definition 3.1.
For L ∈ { C, C ∗ , Q } , we call the summation P z ∈ L φ z an L -shuffle if • ∀ z ∈ L ∀ ( u, v ) ⊆ L , ∃ z ′ ∈ ( u, v ) such that φ z ≦ φ z ′ Shuffles are such that every interval of L contains a cofinal collection of { φ x : x ∈ L } . For example, a simple product L × M is a shuffle. Thereis a similar notion for regular cardinals, which we will need. Note, shufflesextend naturally to quasi order labeled types Φ via the order ≦ defined onthem in section 2.2. Definition 3.2.
Given a regular cardinal κ , we call the type P α<κ φ α κ (resp. κ ∗ ) unbounded if ∀ α ∃ β > α such that φ α ≦ φ β .Note that shuffles are equivalent up to cofinality. That is, if ∀ φ ∈ { φ x : x ∈ L } ∃ ψ ∈ { ψ x : x ∈ L } φ ≦ ψ and vice versa, then P x ∈ L ψ x ≡ P x ∈ L φ x . Theargument for why this is true is outlined in lemma 3.2 and is heavily relianton fact 2.5. 7 efinition 3.3. Given an L sum of labeled Q types Ψ = P x ∈ L Ψ x , U = { Ψ x : x ∈ L } , we say Ψ is universal if Ψ ≧ Φ whenever Φ = P x ∈ L Φ x and V = { Φ x : x ∈ L } is dominated by U i.e ∀ Γ ∈ V ∃ Θ ∈ U Γ ≦ Θ. Fact 3.1.
Every L shuffle for L ∈ { Q , C, C ∗ } , L unbounded sum L ∈ { ω, ω ∗ } of U is universal.Consequently, universal types over L ∈ { ω, ω ∗ , Q , C, C ∗ } are simply L -shuffles or L -unbounded sums respectively up to equivalence.One nice property of shuffles is that every fragmented Aronszajn line canbe embedded into a recursively constructed shuffle. Martinez Ranero origi-nally constructed these lines, though the notion of shuffle was not considered. Fact 3.2. (MA ℵ ) If φ is a fragmented Aronszajn type, there exists an α < ω such that φ ≦ tp( D + α ), where D + α is recursively defined as C -shuffles like so • D +0 = C • D − = C ∗ • For α = β + 1 D + α = C × D − β • For α = β + 1 D − α = C ∗ × D + β • For α limit D + α is a C shuffle of { D + β : β < α }• For α limit D − α is a C ∗ shuffle of { D − β : β < α } Fact 3.3.
Every fragmented Aronszajn line φ of rank α embeds into either D + α or D − α A useful property of shuffles and unbounded sums is that they preservealgebraic indecomposability.
Lemma 3.1.
For L ∈ { C, C ∗ , Q } , if φ is an L shuffle of algebraically inde-composable elements, then φ is algebraically indecomposable.Proof. Suppose S x ∈ L M x is an L shuffle with type φ . Suppose S x ∈ L M x ⊆ A ∪ B where A < B . We may suppose both A and B are nonempty. Then, there isa z ∈ L for which S x We call an L -shuffle P x ∈ L φ x strict ⇐⇒ { φ x : x ∈ L } formsa ≦ -chain. Lemma 3.2. Suppose φ = P x ∈ L ψ x is a strict L shuffle. Then φ ≡ P x ∈ L ϕ x where { ϕ x : x ∈ L } is well ordered under ≦ and has order type or | L | Proof. We can take { ϕ y : y ∈ κ } ⊆ { ψ x : x ∈ L } cofinal with α < β ⇒ ϕ α <ϕ β . There are two cases to consider. Case 1: κ = | L | . Consider the shuffle P y ∈ L ϕ y where each member in thecofinal sequence appears exactly once. It is clear this is a shuffle as everyinterval has size | L | and hence { ϕ y : y ∈ I } is cofinal for any interval I .Consider I a collection of disjoint intervals in L isomorphic to L under theblock sequence order. Let f be such an isomorphism. It is clear now by thecofinality that P x ∈ L ψ x ≦ P y ∈ L ϕ y . For each x ∈ L , we take z ∈ f ( x ) such that ψ x ≦ ϕ z . Doing the same cofinality trick, we can reverse the argument toget P y ∈ L ϕ y ≡ P x ∈ L ψ x . Case 2: κ < | L | . Consequently, κ can embed into L . Consider now ϕ = P α<κ ϕ α . Take an interval partition I of L isomorphic to L . Withineach interval, take κ increasing new disjoint intervals and for the α interval,find a type ψ α ≧ ϕ α . So, for each I ∈ I , we found a type of the form P α<κ ϕ α .Note P α<κ ψ α ≧ P α<κ ϕ α . However, by cofinality of ϕ α , P α<κ ψ α ≡ P α<κ ϕ α . Wehave thus shown that tp( L ) × ϕ ≦ P x ∈ L ψ x . However, for all x ∈ L , ψ x ≦ ϕ ,so tp( L ) × ϕ ≡ P x ∈ L ψ x . Lemma 3.3. Suppose φ = P x ∈ L ψ x is a strict L shuffle with L ∈ { C, C ∗ , Q } and there is an n ∈ ω such that ∀ x ∈ L , ψ x → ( ψ x ) k,n , then φ → ( φ ) k,n roof. Let S x ∈ L M x be such that tp( M x ) = ψ x and is a L shuffle. By theprevious lemma, we may suppose { ψ x : x ∈ L } is well ordered under ≦ .Moreover, by the previous lemma, there are two cases to consider. Either { ψ x : x ∈ L } has order type | L | or 1. Case 1: φ ≡ tp( L ) × ϕ where ϕ is a | L | unbounded sum of types from { ψ x : x ∈ L } i.e the cofinality 1 case. It is clear that ϕ → ( ϕ ) k,n . Then φ → ( φ ) k,n . Case 2: ∀ x, y ∈ L ψ x < ψ y or ψ x > ψ y . Let c : S x ∈ L M x → k with k ≥ n . Foreach x ∈ L , there is an A ⊆ [ n ] k such that ∃ M ′ x ⊆ M x ∩ c − ( A ). Consider c : L → [ n ] k to map each x to some A ∈ [ n ] k such that we can find an M ′ x asabove. Since L is indecomposable, we can find L ′ ⊆ L isomorphic to L andsuch that c ↾ L ′ = A . For each x ∈ L ′ , we find the requisite M ′ x . It is clearthat S x ∈ L ′ M ′ x ⊆ c − ( A ). However, as ψ x is ≦ well ordered of order type | L | and each tp( M ′ x ) are distinct, S x ∈ L ′ M ′ x is an L shuffle. Since tp( M ′ x ) is cofinalin { ψ x : x ∈ L } , tp( S x ∈ L ′ M ′ x ) ≡ φ as desired. Hence, φ → ( φ ) k,n .We now define our class of interest. We cannot isolate just Aronszajn lineswith our method as we want our class to be hereditarily closed. Consequently,we will be interested in the class of Fragmented Aronszajn lines and countableorders. Note, every countable order can be embedded into Q . Definition 3.5. We will define C to be the class of all Fragmented Aronszajnlines and countable orders. We also recursively construct C α for α < ω asfollows. • C = { , } . • φ ∈ C β ⇐⇒ φ = P x ∈ L φ x , φ x ∈ S α<β C α and L ∈ { C, C ∗ , Q } It is clear that D + α , D − α ∈ C α , It is also the case that for any Aronszajn φ ∈ C α , either φ ≦ D + α or φ ≦ D − α . One can prove this by induction. Supposeit is true for all C β for β < α . Take φ ∈ C α . φ = P x ∈ L φ x , L ∈ { C, C ∗ , Q } ,where ∀ x ∈ L ∃ β < α φ x ≦ D + α or D − β . So, φ ≦ P x ∈ L ψ x where ψ x ≡ D + β D − β for some β < α . Note that one of { ψ x : x ∈ L, ∃ β < α ψ x ≡ D + β }{ ψ x : x ∈ L, ∃ β < α ψ x ≡ D − β } is cofinal in { ψ x : x ∈ L } . Consequently, an L shuffle of one of them can embed φ (as shuffles are universal and equivalentup to cofinality). If L = C or C ∗ , we are done. If L = Q , then the rank of φ as an aronszajn line is sup { β : ∃ x ∈ L ψ x ≡ D + β or D − β } ≤ α and so φ ≦ D + α or D − α . Lemma 3.4. C = S α<ω C α .Proof. The inclusion C α ⊆ C for all α < ω is clear. It suffices to show thatand φ ∈ C ⇒ φ ∈ C α for some α < ω . Claim: ∀ φ ∈ S α<ω C α , P x ∈ φ ψ x ∈ S α<ω C α where ψ x ∈ S α<ω C α . Proof. We will show this by way of induction on α where we prove for any α , φ ∈ C α and ∀ x ∈ φ, ψ x ∈ S γ<α C γ P x ∈ φ ψ x ∈ S α<ω C α . It is clearly true for α = 0.Suppose it is true for all α < β . Take φ ∈ C β and ψ x ∈ C β . LetΨ = X x ∈ φ ψ x Without loss of generality, suppose φ ≦ D + β , hence we may assume Ψ = P x ∈ D + β ψ x . But of course, D + β = P y ∈ C ϕ y , where ϕ y ∈ S α<β C α . By our inductionhypothesis, for each y ∈ C , P x ∈ ϕ y ψ y ∈ S α<ω C α . Consequently,Ψ = X y ∈ C X x ∈ ϕ y ψ x ∈ [ α<ω C α To generalize to arbitrary sums of the form φ ∈ S α<ω C α , ∀ x ∈ φ ψ x ∈ S α<ω C α ,one simply needs to find β large enough so that all φ and ψ x embed into D + β .Since S α<ω C α is closed under sums, for any φ ∈ C , we can take α largeenough so that we can find L ⊆ D + α with tp( L ) = φ . Consequently P x ∈ D + α x = φ ∈ S α<ω C α , wher x = 1 ⇐⇒ x ∈ L else x = 0.11e now construct a class of Q labeled lines that will turn out to be thebuilding blocks of C ( Q ), where C ( Q ) are the Q labeled types from C . Thereason for wanting to work with labeled lines is that we can iteratively workwith order types indexed by order types. Definition 3.6. Given a BQO Q , we define H ( Q ) recursively. H ( Q ) = { , q } . φ ∈ H β ( Q ) ⇐⇒ φ is an L shuffle of members U ⊆ S α<β H α ( Q ), L ∈ { Q , C, C ∗ } or is an L unbounded sum for L ∈ { ω, ω ∗ } .In the case Q = { , } , we simply write H as the class is identifiable withthe class of orders with no labels. Proposition 1. Every φ ∈ H is algebraically indecomposable.Proof. We do this by induction. Take φ ∈ H β and suppose the statement istrue for all α < β . By lemma 1.2, the case in which φ = P x ∈ L φ x , ∀ x ∈ L ∃ α < βφ x ∈ H α is an L -shuffle has been accounted for. Suppose instead φ = P k ∈ ω φ k an unbounded sum. The case with ω ∗ is symmetric. Suppose L = S k ∈ ω M k isa realization of φ . Let L = A ∪ B where ∀ x ∈ A y ∈ B x < y and neither isempty. Then there is a m ∈ ω for which S k ∈ ω \ m M k ⊆ B . However, it is clearthat P k ∈ ω φ k ≡ P k ∈ ω \ m φ k . In this section, we show every labeled type in C ( Q ) can be expressed as afinite sum of AI labeled types from H ( Q ). But first, we must show that H ( Q )is a BQO. Definition 3.7. Given a BQO Q , we define Q + to be the disjoint union Q ∪ C . We also define T Q to be the BQO of trees indexed by Q under the ≦ m ordering.Consider the map T : H ( Q ) → T Q + constructed recursively as follows. T (0) = 0, T (1 q ) = 1 q If Ψ ∈ H β ( Q ) \ S α<β H α ( Q ) and is a L-shuffle or L unbounded sum of some types U ⊆ H ( Q ), define T (Ψ) = [ L : { T (Θ) : Θ ∈U } ]. 12 roposition 2. T (Φ) ≦ s T (Ψ) ⇒ Φ ≦ Ψ . We do this by induction on α . Suppose for all α < β , Ψ ∈ H α ⇒ T (Φ) ≦ s T (Ψ) ⇒ Φ ≦ Ψ. It is clear that this is true for H ( Q ). TakeΨ ∈ H β ( Q ). Suppose T (Φ) ≦ s T (Ψ) and is witness by f : T (Φ) → T (Ψ). Case 1: f (root( T (Φ))) = root( T (Ψ)). Since T (Ψ) = [ L : T (Θ) Θ ∈ U ]for some L and some U ⊆ S α<β H α ( Q ), ∃ Θ ∈ H α ( Q ) ∩ U , α < β for which T (Φ) ≦ s T (Θ). By our induction hypothesis, Φ ≦ Θ. However, Ψ is an L shuffle or unbounded sum of U . In particular, Θ ≦ Ψ and we are done bytransitivity. Case 2: f (root( T (Φ))) = root( T (Ψ)). So, T (Ψ) = [ M : T (Θ) Θ ∈ U ]and T (Φ) = [ L : T (Γ)Γ ∈ V ]. It is clear from f that for each Γ ∈ V ∃ Θ ∈ U such that T (Γ) ≦ s T (Θ) ⇒ Γ ≦ Θ by our induction hypothesis.If M = ω (resp. ω ∗ ), then L = ω and Φ = P k ∈ ω Γ k Ψ = P k ∈ ω Θ x . For eachΓ k ∃ Θ m k such that Γ k ≦ Θ m k . Since Φ is an unbounded sum, we can take m k strictly increasing to build an embedding Φ ≦ Ψ.Suppose instead that M ∈ { Q , C, C ∗ } . It follows that L ≦ M . Conse-quently, there is an interval partition I of M that is isomorphic to L . Let ι : L → I be an isomorphism. Since Ψ = P x ∈ M Θ x and Φ = P x ∈ L Γ x and Ψ isa shuffle, for each x ∈ L ∃ z ∈ ι ( x ) such that Γ x ≦ Θ z . Consequently, we canrecursively construct an embedding Φ ≦ Ψ as desired.From the above, it follows that H ( Q ) is BQO for any BQO Q . Lemma 3.5. For any BQO Q , a Q labeled L ∈ { Q , C, C ∗ } can be expressedas a countable sum of universal L types.Proof. We may suppose that for all r ∈ Q , the statement is true for theBQO { q ∈ Q : r (cid:20) q } . Let ( L, l ) be a Q labeling. Consider the equivalencerelation ∼ on L , x ∼ x , x ∼ y ⇐⇒ ∀ ( u, v ) ⊆ ( x, y ), ( u, v ) is a countablesum of universal L types. Each class is a countable sum of universal L types as ( x, y ) ≡ L and has countable cofinality. So, suppose otherwise.Take X, Y ∈ L/ ∼ . It is clear that ( X, Y ) ≡ L . Moreover, for all q ∈ Q , ∃ Z ∈ ( X, Y ) and z ∈ Z such that l ( z ) ≧ q . If not, then any z ∈ ∪ Z ∈ ( X,Y ) Z q (cid:20) l ( z ) and we are in our base case. Hence, X = Y and wehave a contradiction. It follows then that our original labeled line ( L, l ) wasuniversal. Lemma 3.6. Given χ ∈ H ( H ( Q )) , χ = ( X, l ) , the Q labeled order order χ = P x ∈ X l ( x ) is in H ( Q ) Proof. This can be done via a simple induction argument. It is clear thatstatement is true for H ( H ( Q )). If it is true of H α ( H ( Q )) for every α < β ,then converting M to shuffle or unbounded sum of members from H α ( H ( Q )),we can imply our induction hypothesis and conclude as H ( Q ) is closed undershuffles and unbounded sums. Theorem 3.7. Let Q be a BQO. Any Q type Φ ∈ C α ( Q ) can be written as afinite sum of members from H ( Q ) .Proof. We do this by induction on β . It is definitely true for β = 0. Supposethe statement is true for all α < β . If the base of Φ is countable, we are doneby Laver’s theorem [6] so suppose otherwise. By lemma 3.5, Φ = P x ∈ L Φ x ,bs(Φ x ) ∈ S α<β C α , L ∈ { Q , ω, ω ∗ , C, C ∗ } . Case 1: L ∈ { ω, ω ∗ } . Then by our induction hypothesis, Φ = P k ∈ L (Ψ k, + ... + Ψ k,r k ) where Ψ k,r k ∈ H ( Q ) and r k is a sequence of integers. But then,up to reorganization, Φ = P k ∈ ω Θ k , where Θ k ∈ H ( Q ). Since H ( Q ) is BQO,there is a minimal m for which ∀ k > m , Θ m (cid:20) Θ k . If not, we can constructa nowhere increasing subsequence Θ m k , contradicting the BQO assumption.But then, P k ∈ ω \{ ,...,m } Θ k is an unbounded sum of members from H ( Q ) andhence, a member of H ( Q ) itself. But then, Φ = Θ + ... + Θ m + P k ∈ ω \{ ,...,m } ,a finite sum of members from H ( Q ) Case 2: L ∈ { Q , C, C ∗ } . By our induction hypothesis, Φ = P x ∈ L (Φ x, + .., Φ k x )where k x is an integer. As L ≡ L , up to reorganization, we may supposethat Φ = P x ∈ L Φ x where each Φ x is in H α ( Q ) for some α < β . Considerthe χ ∈ H ( H ( Q )), χ = ( L, l ), l ( x ) = Φ x . By lemma 3.5, χ = P x ∈ S Ψ ′ x ,14 ′ x ∈ H ( Q ) and S countable. By Laver’s theorem, tp( S ) = Ω + ... + Ω k where Ω i ∈ H ( { , } ). So then, χ = χ + ... + χ k where each χ i ∈ H ( H ( Q )).But then, Φ ≡ χ = χ + ... + χ k and we are done by lemma 3.6. Corollary 3.7.1. (PFA) Given a BQO Q , the class C ( Q ) is BQO. Corollary 3.7.2. (PFA) The class of all Aronszajn lines is BQO. Corollary 3.7.3. In conjunction with Lavers’ result from [6], the class ofall Aronszajn lines and σ -scattered orders closed under summations over oneanother is BQO. Being able to decompose Aronszajn lines into a finite sum of AI types is aquite powerful result. In particular, it means that our shuffles could havebeen strict with no change to the class H . In this subsection, we will useour newly found results to show that every member in H could have beencoded by a finite tree labeled with AI types. First, we must define the classof trees. Definition 3.8. Consider the class U ⊆ F T H defined recursively as follows. U = {∅ , } where 1 is the one node tree indexed by 1 and ( T, l ) ∈ U α ifand only if one of the following holds • ( T, l ) ∈ U β for some β < α • ∃ L ∈ { Q , C, C ∗ } , if ∀ x ∈ L , ( T, l x ) ∈ U β for some β < α and ∀ x, y, z ∈ L , ∃ u ∈ ( x, y ) such that ( T, l z ) ≦ I ( T, l x ), then P x ∈ L ( T, l x ) = ( T, l ) ∈ U . • ( T, l ) = P n ∈ ω ( T, l n and ∀ n ∈ ω ∃ β < α ( T, l n ) ∈ U β and n < m ⇒ ( T, l n ) ≦ I ( T, l m ). • ( T, l ) = [ ω ; ( T , l ) , ..., ( T n , l n )] (resp. [ ω ∗ ; ( T , l ) , ..., ( T n , l n )]) with ( T , l ) , ..., ( T n , l n ) ∈U β for some β < α and ( T i , l i ) = P n ∈ ω ( T i , l i,n ) (resp. P n ∈ ω ( T i , l i,n )) and for i < k ( T i , l i,n ) ≤ I ( T k , l k,n ). Definition 3.9. Given ( T, l ) ∈ U , we assign a linear order ( T, l ) in H recur-sively as follows. 15 = 1 • ∅ = 0 • If ( T, l ) = P x ∈ L ( T, l x ) for L ∈ { Q , C, C ∗ , ω, ω ∗ } , then ( T, l ) = P x ∈ L ( T, l x ) • If ( T, l ) = [ L ; ( T , l ) , ..., ( T n , l n )] for L ∈ { ω, ω ∗ } , ( T, l ) = P x ∈ L ( T , l ,x )+ ... + ( T n , l n,x )We let U = S α ∈ Ord U α From the set up, it should be clear that our proof is going to require aninduction proof. The following lemma will allow us to easily compare L and M shuffles to one another when L and M are independent with respect to ≦ . Lemma 3.8. Suppose L , L ∈ { ω, ω ∗ , C, C ∗ , Q } and P x ∈ L φ x ≦ P x ∈ L ψ x whereboth sums are shuffles of AI objects. Then, at least one of the following mustoccur. • ∃ z ∈ L for which P x ∈ L φ x ≦ ψ z • L ≦ L Proof. Take an embedding f : P x ∈ L M x ≦ P x ∈ L N x . The result is trivial if L , L ∈ { ω, ω ∗ } so we ignore these cases. Note that for a given M x , f [ M x ]cannot be cofinal. Consider the mapping g from L into bounded sets of L that maps x to the set { y ∈ L : P y ∩ f [ M x ] = ∅} . Note that g hasthe property that x < L y ⇒ g ( x ) < g ( y ) in the block sequence order or g ( x ) ∩ g ( y ) is a singleton and max g ( x ) = min g ( y ). Consider the equivalencerelation ∼ defined as follows. ∀ x, y ∈ L ∀ u, v ∈ ( x, y ) , u < v, max g ( u ) = min g ( v ) ⇒ x ∼ y ∀ x ∈ L x ∼ xx ∼ y ⇒ y ∼ x It is clear that ∼ is an equivalence relation with convex equivalence classes.There are two cases to consider. The first is that ∃ X ∈ L / ∼ such that16 ≡ L . In this case, ∀ u ∈ X , g ( u ) is a singleton and P x ∈ X M x ≡ P x ∈ L M x onthe account that X is convex and the sum was a shuffle. But then, f ↾ P x ∈ X M x has range in N g ( u ) for some u ∈ X and we are done.Suppose instead that no X ∈ L / ∼ is equivalent to L . In this case, { g ( x ) : x ∈ L } forms a block sequence. Taking a selector s : L → L , s ( x ) ∈ g ( x ), we have shown L ≦ L .In our main proof, we will see how this extends to trees under the ≦ m ordering. Given that the trees our finite, we will get a similar result by ap-plying the above lemma a finite number of times.Akin to proposition 2, we must hope that two trees being comparable withrespect to ≦ m gives us some tangible information about the orders they code.Fortunately, this is true. This requires an exhaustive case analysis. Proposition 3. If ( S, l ) ≦ m ( T, m ) , then ( S, l ) ≦ ( T, m ) .Proof. We work on induction on U α . Suppose for all ( T ′ , m ′ ) ∈ U α for α < β , ( S, l ) ≦ m ( T ′ , m ′ ) ⇒ ( S, l ) ≦ ( T ′ , m ′ ). Take ( T, m ) ∈ U β . Sup-pose ( S, l ) ≦ m ( T, m ) ⇒ ( S, l ) ≦ ( T, m ) for ( S, l ) ∈ U α for all α < γ . Take( S, l ) ≦ m ( T, m ) with ( S, l ) ∈ U γ . We may also assume the result is truefor all ( T ′ , m ′ ) < ( T, m ) and ( S ′ , l ′ ) < m ( S, l ) by BQO induction. Note, thestatement is trivial if either S or T is a singleton. Case 1a: ( T, m ) = [ ω ; ( T , m ) , ..., ( T n , ..., m n )] and ( S, l ) = [ ω ; ( S , l ) , ..., ( S k , l k )].Then, ∀ i ∃ j such that ( S i , l i ) ≦ m ( T j , m j ). Suppose it is the case that( S i , l i ) and ( S k , l k ) both embed into ( T j , m j ). As ( T j , m j ) = P x ∈ ω ( T j , m j,x ) ≡ P x ∈ ω even ( T j , m j,x ) ≡ P x ∈ ω odd ( T j , m j,x ), we can embed both ( S i , l i ) and ( S k , l k )simultaneously. Consequently, one can further embed ( S, l ) into ( T, m ). Thecase of ( S, l ) = [ ω ∗ ; ( S , l ) , ..., ( S k , l k )] is symmetric. Case 1b: ( T, m ) = [ ω ; ( T , m ) , ..., ( T n , ..., m n )] and ( S, l ) = P x ∈ L ( S, l x ) forsome L ∈ { Q , C, C ∗ , ω, ω ∗ } . Then it must follow that ( S, l ) ≤ m ( T i , m i ) forsome i and we are done. 17 ase 2a: ( T, m ) = P x ∈ M ( T, m x ) for M ∈ { ω, ω ∗ , C, C ∗ , Q } and for each x ∈ M , ( T, m x ) ∈ U α for some α < β . ( S, l ) = P x ∈ L ( S, l x ) for some L ∈{ C, C ∗ , ω, ω ∗ , Q } . Let f : S → T be an embedding witnessing ( S, l ) ≦ m ( T, m ). If L (cid:20) M , then by lemma 2.4, for each s ∈ S , ∃ z s ∈ M suchthat f ( l ( s )) ≦ m z s ( f ( s )). Since { ( T, m z ) z ∈ M } is linearly ordered un-der ≦ I , there is some z ∈ M (in particular, the max z s under the order z s ≤ z t ⇐⇒ ( T, m z s ) ≦ I ( T, m z t )) for which ( S, l ) ≦ m ( T, m z ). But then,by our induction hypothesis, ( S, l ) ≦ ( T, m z ) ≦ ( T, m ).Suppose instead that L ≦ M . The result is trivial if M ∈ { ω, ω ∗ } so wesuppose otherwise. Take I a collection of intervals in M that is isomorphicto M under the block ordering. Take an embedding ι : L → I . By ourinduction hypothesis, for each x ∈ L , ∃ f x : ( S, l x ) → P y ∈ ι ( x ) ( T, m y ) ≡ ( T, m ).So then, f = S x ∈ L f x witnesses ( S, l ) ≦ P x ∈ L P y ∈ ι ( x ) ( T, m y ) ≡ ( T, m ). Case 2b: ( T, m ) = P x ∈ M ( T, m x ) for M ∈ { ω, ω ∗ , C, C ∗ , Q } and for each x ∈ M , ( T, m x ) ∈ U α for some α < β . ( S, l ) = [ L, ( S , l ) , ..., ( S, l n )] L ∈ { ω, ω ∗ } . The cases for M ∈ { ω, ω ∗ } is the simplest and was doneexplicitly by Laver. For each i ∈ { , ..., n } , ( S i , l i ) ≦ m ( T, m ). In particular,by our induction hypothesis, ( S i , l ij ) ≦ ( T, m ). Take a collection of inter-vals I in M block isomorphic to n × L . Take ι : n × L → I , and for each( i, j ) ∈ n × L , take an embedding f ij : ( S i , l ij ) → ι ( i, j ). Then, S ( i,j ) ∈ n × L f ij witnesses an embedding of ( S, l ) = P j ∈ L n P i =1 ( S i , l i,j ) into ( T, m ). Lemma 3.9. If P x ∈ L φ x = Φ is a shuffle with φ x < Φ for L ∈ { C, C ∗ } , then Φ ≡ P x ∈ L ψ x where each ψ x is AI and { ψ x : x ∈ L } is totally ordered.Proof. By Theorem 3.7, we may assume each φ x is AI. Ordering φ x via φ α , α < ω , we can instead analyze the sum P α ∈ ω φ α . There exists ψ α < P α ∈ ω φ α such that α < β ⇒ ψ α ≦ ψ β and ∀ α ∃ β for which φ α ≦ ψ β . It becomes clearthat an L shuffle of ψ α will suffice. Proposition 4. For each φ ∈ H ∃ ( T, l ) ∈ U such that ( T, l ) ≡ U roof. Suppose the result is true for all ψ < φ . Suppose φ = P x ∈ L ψ x L ∈ { C, C ∗ , Q , ω, ω ∗ } a shuffle. First suppose that { ψ x : x ∈ L } is or can betotally ordered under a change like in lemma 2.6. For each x ∈ L , find ( T x , l x )with ( T x , l x ) = ψ x . If | L | > ℵ , we can suppose T x = T y for all x, y ∈ L . Inthis instance, we may also use the BQO property to further move down to asubsequence with { ( T x , l x ) : x ∈ L } linearly ordered under ≦ I and we’d bedone. Suppose instead that | L | = ℵ . By Laver’s covering theorem, we maysuppose ( T x , l x ) have at most n treetops, have height bounded by n , and is ≤ m increasing. But then, there are only finitely many types of trees, meaningwe can go down to a class of trees ( T x , l x ) with T x = T y for all x, y ∈ L . Butthen, again we may assume the trees are ≤ I ascending like in the previouscase, so that ( T, l ) = P x ∈ L ( T x , l x ) does the job.If { ψ x : x ∈ L } cannot be totally ordered under ≦ , then L ∈ { ω, ω ∗ } .Then we may suppose φ = P x ∈ L n P i =1 ψ x,i , where for a fixed i , { ψ x,i : x ∈ L } istotally ordered. Applying the previous case, we get ( T i , l i ) = P x ∈ L ( T i , l i,x ) with( T i , l i,x ) = ψ i,x . But then, ( T, l ) = [ L : ( T , l ) , ..., ( T n , l n )] does the trick. Theorem 3.10. If ( T, l ) ∈ U has n treetops, then ( T, l ) has Ramsey degreebounded by n .Proof. We do this by induction on U α . It is true for α = 0. Suppose it istrue for all ( T, l ) ∈ U α for α < β . Let ( T, l ) ∈ U β . If ( T, l ) is a shuffle, we aredone by lemma 1.3. Suppose instead that ( T, l ) = [ ω, ( T , l ) , ..., ( T r , l r )] with( T i , l i ) ∈ U α for some α < β . Each ( T i , l i ) = P k ∈ ω ( T i , l i,k ). Note, ( T i , l i ) has n i tree tops and hence Ramsey degree bounded by n i . ( T, l ) has n = r P i =1 n i manytree tops ( T, l ) = P k ∈ ω r P i =1 ( T i , l i,k ). Let M = S k ∈ ω ( r S i =1 M i,k ) be a realization ofthis type. For i ≤ r , we define M i = S k ∈ ω M i,k . Note, each M i has Ramseydegree n i and M i forms a partition of M . Consequently, M has Ramseydegree bounded by n . Corollary 3.10.1. (PFA) For any Aronszajn type φ , there is an n ∈ ω suchthat ∀ k ∈ ω φ → ( φ ) k,n . roof. The result is true if φ is not fragmented so suppose otherwise and let φ ∈ C . By theorem 3.7, φ = φ + ... + φ k where each φ i ∈ H . By proposition4, for each φ i there is ( T i , l i ) such that ( T i , l i ) ≡ φ i . By theorem 3.10, each( T i , l i ) has finite Ramsey degree. Consequently, φ has finite Ramsey degreeby lemma 2.1. 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