aa r X i v : . [ m a t h . L O ] J a n Borel Colouring Bad Sequences
Keegan Dasilva Barbosa
Abstract
Every better quasi-order codifies a Borel graph that does not con-tain a copy of the shift graph. It is known that there is a better quasi-order that codes a Borel graph with infinite Borel chromatic number,though one has yet to be explicitly constructed. In this paper, we showthat examples cannot be constructed via standard methods. More-over, we show that most of the known better quasi-orders are non-examples, suggesting there is still a class of better quasi-orders withinteresting combinatorial properties who’s elements/members still re-main unknown.
In [8], it was discovered that there is a graph with uncountable Borel chro-matic number which is minimal with respect to Borel graph homomorphism.It was then questioned whether or not the shift graph was minimal in thisregard in the class of Borel graphs generated by a single Borel function. In-terestingly, Pequignot showed in [17] that the shift graph contained a Borelsubgraph with infinite Borel chromatic number, but could not embed theshift graph in a Borel manner. These graphs were codified by better quasi-orders (BQO). Further work was done by Todorcevic and Vidnyanszky, andit was further proved that no finite basis could exist, let alone a singularminimal element [19]. However, there is a deep question that still remainsunanswered in [17]. While it is known a BQO can codify a graph with infiniteBorel chromatic number, by the nature of the proof relying on a complexityargument of Marcone [13],a concrete BQO has yet to be identified or explic-itly constructed. There were some conjectures made by Pequignot. In thispaper, we show that these BQOs fail to codify a graph with infinite Borelchromatic number. Moreover, we show that a BQO recursively constructed1rom simpler BQOs by the classical means of labeling trees or linear orders[10] [11] [18] also fails to produce graphs with infinite Borel chromatic num-ber. As a consequence, even any countable collection of σ -scattered linearorders under the embedding relation fails to be complex enough to code agraph with infinite Borel chromatic number. This suggests there is still awealth of BQOs with strong indecomposability properties that have yet tobe explicitly constructed.This paper will be split into two sections excluding introduction and ac-knowledgements. Section 2 will be focused on the necessary backgroundinformation required to understand the problem. This includes the basicsof Borel graph combinatorics, BQO theory, and the primary tools and tech-niques used in these fields. It is suggested a reader skips through the partsof this section they are familiar with. If one wants a deeper understandingof these materials, the author suggests any of the following texts [5] [7] [9][14][12]. However, section 2.5 should not be skipped. It contains importantdefinitions specialized for this problem, as well as a very important lemma.Section 3 is split into three components. Section 3.1 is where we will intro-duce the colouring algorithm that will allow us to effectively Borel 3-colourgraphs by splitting them into homogeneous pieces that are easy to handle.Section 3.2 will be where we prove that if Q codes a Borel 3-colourable graph(we call such Q thin ), then so does Q <ω under the Higman order [4]. Conse-quently, the same will be true of finite trees labelled by members of Q , F T Q ,under the standard orders [10] [11] [18]. Finally, section 3.3 will be where wejump from finite to infinite, and show that if Q is thin, so is the class of Q labeled σ -scattered linear orders under ≦ emb . Definition 2.1.
We call a topological space X Polish if its topology is com-plete, separable, and metrizable.Not much on the finer combinatorics of Polish spaces will be relevantfor us. We’ll mostly be interested in the space of countable sequences of acountable set under the topology of pointwise convergence. However, thereader is highly encouraged to read [5] for more on the subject.2 efinition 2.2.
Given a set A , [ A ] ω = { B ⊆ A : | B | = ω } . Fact 2.1.
Given a countable set A , A ω under the topology of pointwiseconvergence is Polish.Note that if A is countable, [ A ] ω is also Polish with topology determinedby X n → X if and only if ∞ T n =1 ∞ S j = n X j = X . Alternatively, after well ordering A , one can identify members from [ A ] ω with strictly increasing sequences in A ω with the subspace topology. These definitions are equivalent. Fact 2.2.
Given a set A , we define A <ω to be the set of functions withdomain { , ..., n } for some n ∈ ω and codomain A .Elements in this set are finite sequences of members from A . In fact,considering the members as finite sequences will often be of more use to usthan thinking of them as functions. Definition 2.3.
Given a set A , we define the set [ A ] <ω = { B ⊆ A : | B | < ω } . Definition 2.4.
Given two Polish spaces V and V , we say a function f : V → V is Baire class 1 if and only if there is a sequence of continuousfunctions f k : V → V , f k converges to f pointwise.All Baire class 1 functions are Borel. Moreover, being Baire class 1 guar-antees the preimage of every open set is G δ (countable intersection of opensets). One can check [6] for more on Baire class 1 functions. We will needthat these functions are Borel, and not much else. Their main appearance isin the construction of the ∂ ∞ operation in section 3.2. Definition 2.5. A Borel graph G is a pair ( V, E ), where V is a Polish spaceand E ⊆ [ V ] is Borel.We give [ V ] the topology it inherits when viewed as a subset of V .Since we are working in the context of descriptive set theory, all propertiesthat we will be discussing from graph theory need to be Borel definable. Bythis, we mean that we will not be interested in graph homomorphisms, butrather Borel graph homomorphisms. Nor will we work with the standardchromatic number. While it may seem as though this is a small requirement,it is actually rather major. Some graphs that are bipartite in the classic sensemay have large Borel chromatic number. For more on this, see [8].3 efinition 2.6. Given Borel graphs G = ( V , E ) and G = ( V , E ), a Borel graph homomorphism is a Borel map f : V → V with the propertythat ∀ x, y ∈ V , xE y ⇒ f ( x ) E f ( y ). If such a homomorphism exists, wewrite G (cid:22) B G . Definition 2.7.
Given a Borel graph G = ( V, E ), we define the
Borel chro-matic number of G to be χ B ( G ) = min {| Y | : Y Polish , G (cid:22) B ( Y, [ Y ] ) } Note that the requirement that the graph Y in the above definition bePolish means that the only possibilities for χ B ( G ) are { , , , , ..., ℵ , ℵ } .We will be studying graphs whose edge set is determined by Borel functions.The range of plausible Borel chromatic numbers these graphs can achieve isfinite. In particular, the Borel chromatic number can either be 1, 2, 3, or ℵ . Definition 2.8.
We define s : 2 ω → ω via ∀ k ∈ ω, s ( x )( k ) = x ( k + 1). Definition 2.9.
We say a Borel graph G = ( V, E f ) is generated by the Borelfunction f on V if f : V → V Borel and ∀ x, y ∈ V, xE f y ⇐⇒ ( x = y ) and ( f ( x ) = y or f ( y ) = x ) Fact 2.3.
Given a Borel graph G generated by a Borel function f on a Polishspace V , the following are equivalent. • χ B ( G ) ≤ • G (cid:22) B (2 ω , E s ) • ∃ A ⊆ X Borel and f independent such that ∀ x ∈ X ∃ k ∈ ω , f k ( x ) ∈ A (such a set is called forward recurrent ) [14]The universality property of (2 ω , E s ) is rather instrumental for our colour-ing algorithm. The goal of the algorithm is to embed as much as we can into(2 ω , E s ) in a Borel fashion, then work with the homogeneous remainder. Thethird property of forward recurrence will also be relevant. Consequently, fa-miliarity with the above fact will be of high importance. Note, not everyBorel graph generated by a function needs finite Borel chromatic number.The shift graph is a counterexample. 4 efinition 2.10. We call the graph ([ ω ] ω , S ) where S ( A ) = A \ min A the shift graph . Fact 2.4 (Galvin-Prikry) . Given a k ∈ ω and a Borel colouring c : [ ω ] ω →{ , ..., k } , there is an A ∈ [ ω ] ω and i ∈ { , ..., k } such that [ A ] ω ⊆ c − ( i ). Fact 2.5. χ B (([ ω ] ω , S )) = ℵ . Proof.
Simple application of the Galvin-Prikry theorem.
Definition 2.11.
Let f : X → X and g : Y → Y be functions. We call themapping h : X → Y a factor map if ∀ x ∈ X , h ( f ( x )) = g ( h ( x )). Lemma 2.1.
Let X and Y be Polish spaces. Let f : X → X and g : Y → Y be Borel with no fixed point. If h : X → Y is a Borel factor map, then χ B (( X, E f )) ≤ χ B (( X, E g )) .Proof. It suffices to show that (
X, E f ) (cid:22) B ( Y, E g ). The factor map h : X → Y is a graph homomorphism. To see this, take x, y ∈ X with xE f y . Since f has no fixed point, either f ( x ) = y or f ( y ) = x . Without loss of generality,suppose f ( x ) = y . It follows that h ( y ) = h ( f ( x )) = g ( h ( x )). Since g has nofixed point, h ( x ) = h ( y ) and so it must be the case that h ( x ) E g h ( y ). Hence, h is a Borel graph homomorphism.Most of the graphs we will be interested in will have no fixed point. Note,the shift operation S on [ ω ] ω has no fixed point. Definition 2.12. A quasi-order is a pair ( Q, ≦ Q ) where the relation ≦ Q on Q is reflexive and transitive. Definition 2.13.
Given two quasi-orders Q and Q , we assign an order ≦ Q ∪ Q to the disjoint union where p ≦ Q ∪ Q q ⇐⇒ ∃ i ∈ { , } , p, q ∈ Q i and p ≦ Q i q . Definition 2.14.
We say a quasi-order ( Q, ≦ Q ) is a better-quasi-order (BQO)if when Q is endowed with the discrete topology, any Borel (equiv. continu-ous) map f : [ ω ] ω → Q , ∃ A ∈ [ ω ] ω , ∀ B ∈ [ A ] ω , f ( B ) ≦ Q f ( S ( B )).5 weaker notion is that of a well quasi-order . They’re much easier towork with, but do not code the type of graphs we are interested in. They’realso not closed under as many operations as BQOs are. Note also the strongtie between the Galvin-Prikry theorem and the definition of BQO. Definition 2.15.
Given a BQO Q , we order R = Q <ω via t ≦ R t ⇐⇒∃ h : dom( t ) → dom( t ) order preserving such that t ( n ) ≦ t ( h ( n )). Wecall this order the Higman order . Fact 2.6. If Q is a BQO, then Q <ω is BQO under the Higman order. Definition 2.16. A tree is a pair ( T, ≤ T ) where ≤ T is a partial order withthe property that ∀ t ∈ T , { v ∈ T : v ≤ T } is well ordered under ≤ T , and T has a ≤ T minimal element called the root . Definition 2.17.
Given a set Q , we define F T Q to be the class of all pairs( T, l ) where T is a finite tree and l : T → Q . We call members of this class finite labeled trees and refer to the function l as a labeling . Definition 2.18.
Given a set Q , we define T Q to be the class of all pairs( T, l ) where T is a tree of height ≤ ω and l : T → Q . We call members ofthis class labeled trees and refer to the function l as a labeling . Definition 2.19.
Let Q be a quasi-order. We define two quasi-orderings, ≦ and ≦ m on T Q via • ( T , l ) ≦ ( T , l ) ⇐⇒ ∃ f : T → T , ∀ x, y ∈ T , x < T y ⇒ f ( x ) < T f ( y ) and l ( x ) ≦ Q l ( f ( x )). • ( T , l ) ≦ m ( T , l ) ⇐⇒ ∃ f : T → T , ∀ x, y ∈ T , x ≤ T y ⇒ f ( x ) ≤ T f ( y ) and l ( x ) ≦ Q l ( f ( x )).The main distinction between ≦ and ≦ m is ≦ requires the function f be injective. For ≦ m , the function need only preserve the tree ordering. Fact 2.7. (Laver) If Q is BQO, then T Q and F T Q are BQO under ≦ and ≦ m .In the next subsection, we will briefly highlight results from Lavers workon σ -scattered orders [10] [11]. Codifying objects on labeled trees is one ofthe most natural ways to prove an order is a BQO. For example, considerPouzet’s theorem [18]. 6 .4 Basics of σ -scattered Orders Definition 2.20.
Given linear orders L and L , we say L ≦ emb L if thereexists a function f : L → L , ∀ x, y ∈ L , x < L y ⇒ f ( x ) < L f ( y ). We callsuch a map an embedding . If L ≦ emb L , we say L embeds a copy of L . Definition 2.21.
Given a linear order L , we define its reverse order L ∗ tobe the pair ( L, < L ∗ ) with x < L ∗ y ⇐⇒ x < L y . Definition 2.22.
A linear order is called scattered if it does not embed acopy of the rationals Q . Definition 2.23.
A cardinal κ is called regular if ever cofinal subset hascardinality κ .For more on the basics of cardinals, see [9]. Definition 2.24.
We define RC to be the class of regular cardinals.Note that every regular cardinal is scattered. Moreover, RC is a BQOunder ≦ emb . Definition 2.25.
A linear order L is called σ -scattered if it can be expressedas a countable union L = ∞ S n =1 L n such that L n are scattered. Definition 2.26. A Q - labeled order is a pair ( L, l ), where L is a linear orderand l : L → Q . We call such an l a labeling. Definition 2.27.
Given a set Q and a cardinal κ , Q κ is the set of labeledordinals α , where α < κ . Definition 2.28.
Let Q be a quasi order. Given two labeled orders ( L , l )and ( L , l ), we say ( L , l ) ≦ emb ( L , l ) if there is an embedding f : L → L with the property that ∀ x ∈ L l ( x ) ≦ Q l ( f ( x )). Definition 2.29.
Given a quasi-order Q , we define the class C ( Q ) to be theclass of Q -labeled σ -scattered linear orders, ordered under ≦ emb . Fact 2.8.
Given α, β ∈ RC uncountable, there is a unique linear order η α,β that is ≦ emb maximal over the class of σ -scattered orders L that do not embed α ∗ or β . 7ausdorff showed that regular cardinals served as the building blocks ofthe scattered linear orders. Laver showed via similar means that regularcardinals paired with the above orders serve as the building blocks for all σ -scattered orders. It is also interesting to not that η ω ,ω is the rationals. Definition 2.30.
Given a quasi order Q , we define Q + to be the disjointunion Q ∪ { κ : κ ∈ RC } ∪ { κ ∗ : κ ∈ RC } ∪ { η α,β : α, β ∈ RC } . Fact 2.9 (Laver) . Given a better quasi order Q , there is a class of σ -scatteredlinear orders H ( Q ) and a mapping J : H ( Q ) → T Q + with the property that ∀ ( L , l ) , ( L , l ) ∈ H ( Q ), ( L , l ) (cid:20) emb ( L , l ) ⇒ J (( L , l )) (cid:20) J (( L , l )). Fact 2.10 (Laver) . There is a map J : C ( Q ) → H ( Q ) <ω such that ( L , l ) (cid:20) emb ( L , l ) ⇒ J (( L , l )) (cid:20) H ( Q ) <ω J (( L , l )).These facts imply that if Q is BQO, then so is the class of Q -labeled σ -scattered linear orders under ≦ emb . Definition 2.31.
Given a better quasi order Q , we define ~Q = { X ∈ Q ω : ∀ k ∈ ω, X ( k ) (cid:20) Q X ( k +1) } . We often conflate ~Q with the shift graph ( ~Q, S ).One property these graphs have is they do not homomorphically embedthe shift graph ([ ω ] ω , S ) as a consequence of Q being BQO [17]. Pequignotbegan a search for a BQO Q with the property that χ B ( ~Q ) = ℵ . Since we’realso interested in this property, we will give it a name. Definition 2.32. If χ B ( ~Q ) ≤
3, we call Q thin . If Q is not thin, we call it thick .We will later show that the name “thick” is rather appropriate when wediscuss a corollary to the colouring algorithm. Lemma 2.2.
Suppose Q and Q are BQO and there is a map f : Q → Q with q (cid:20) Q p ⇒ f ( q ) (cid:20) Q f ( p ) . Then Q thin ⇒ Q thin.Proof. Consider the map ~f : ~Q → ~Q given by ~f ( X )( k ) = f ( X ( k )) for all k ∈ ω . Note, for all k ∈ ω , X ( k ) (cid:20) Q X ( k + 1) ⇒ f ( X ( k )) (cid:20) Q f ( X ( k + 1))so this mapping is well defined. It is also continuous as X n → X point-wisein ~Q will imply that ~f ( X n ) → ~f ( X ) point-wise. Moreover, it is easy tocheck that ~f is a factor map. It follows that ~Q (cid:22) B ~Q .8n immediate corollary to the above is the unsurprising fact that the thinproperty is hereditary. That is to say, if Q is thin and Q ′ ⊆ Q is given theorder it inherits from Q , then Q ′ is also thin. More interestingly, there hasbeen historically many instances where maps of this sort are used to provean ordering is BQO. For example, consider the previous facts of Laver fromthe last subsection. It also allows us to reasonably extend the notion of thinto larger BQO’s that may not be sets, but rather classes. Definition 2.33.
Given a BQO Q , we say Q is thin if and only if ∀ Q ′ ∈ [ Q ] ω , Q ′ is thin.Unless we are talking about a concrete BQO, we will always assume athick or thin BQO is countable when we work in the abstract. We begin this section by proving the colouring algorithm.
Lemma 3.1 (colouring algorithm) . Let Q be a better quasi order and Φ be arelation of arity n on Q . ~Q is thin if and only if the following two propertieshold • { X ∈ ~Q : ∀ k ∈ ω, Φ( X ( k ) , ..., X ( n + k )) } is Borel -colourable. • { X ∈ ~Q : ∀ k ∈ ω, ¬ Φ( X ( k ) , ..., X ( n + k )) } is Borel -colourable.Proof. It is clear that if ~Q is thin, then the above two sets are Borel threecolourable as they are both proper induced subgraphs of ~Q . For this reason,we need only show the other direction.Consider the function f : ~Q → ω via f ( X )( k ) = 1 ⇐⇒ Φ( X ( k ) , ..., X ( k + n ))This function is continuous and is a factor map eg. f ( S ( X )) = s ( f ( X )).Moreover, if we let Y = { x ∈ ω : ∀ k ∈ ω ∃ m ≥ k, x ( m ) = x ( k ) } (the set ofsequences which are never eventually constant), then we see f − ( Y ) ⊆ ~Q isBorel, closed under S and is Borel three colourable by lemma 2.1. By fact9.3, there is an A ⊆ f − ( Y ) that is forward recurrent and S independent. ~Q \ f − ( Y ) is the set of all sequences X ∈ ~Q that satisfy one of the followingconditions. • ∃ m ∈ ω , ∀ j ≥ m , S j ( X ) ∈ { X ∈ ~Q : ∀ k ∈ ω, Φ( X ( k ) , ..., X ( n + k )) } • ∃ m ∈ ω , ∀ j ≥ m , S j ( X ) ∈ { X ∈ ~Q : ∀ k ∈ ω, Φ( X ( k ) , ..., X ( n + k )) } By our hypothesis, { X ∈ ~Q : ∀ k ∈ ω Φ( X ( k ) , ..., X ( n + k )) } and { X ∈ ~Q : ∀ k ∈ ω ¬ Φ( X ( k ) , ..., X ( n + k )) } admit forward recurrent S independent sets A and A respectively. Since every X ∈ ~Q is either in f − ( Y ) or eventuallyin A or A , A ∪ A ∪ A is a witness to χ B ( ~Q ) ≤ S independent forward recurrent set. Note, independence is a consequence ofeach A i belonging to disjoint S closed sets.Here are two simple applications of the lemma that will be of use later. Lemma 3.2. If Q and Q are thin, then the disjoint union R = Q ∪ Q isthin.Proof. Consider the unary relation Φ on R given by Φ( q ) ⇐⇒ q ∈ Q . Byour colouring algorithm, it suffices to consider whether or not the subgraphsinduced by the following subsets are Borel 3-colourable. B = { X ∈ R : ∀ k ∈ ω, X ( k ) ∈ Q } B = { X ∈ R : ∀ k ∈ ω, X ( k ) ∈ Q } Notice however, that the first is simply ~Q and the second is ~Q . As both Q and Q are thin, we are done.Note, this implies that finite disjoint unions, and potentially non-disjointunions, of thin BQOs are thin as well. Lemma 3.3. If Q and Q are thin, then R = Q × Q is thin.Proof. Consider the binary relation Φ on R given by Φ(( p , p ) , ( q , q )) ⇐⇒ p ≦ Q q . Let π : R → Q and π : R → Q be canonical projection maps( π i ( p , p ) = p i ). By our colouring algorithm, we need only consider thecolourability of the subgraphs generated by the sets B = { X ∈ ~R : ∀ k ∈ ω Φ( X ( k ) , X ( k + 1)) } B = { X ∈ ~R : ∀ k ∈ ω ¬ Φ( X ( k ) , X ( k + 1)) } p , q ) (cid:20) R ( p , q ), then either p (cid:20) Q q or p (cid:20) Q q . Con-sequently, ¬ Φ(( p , p ) , ( q , q )) and ( p , p ) (cid:20) R ( q , q ) ⇒ p (cid:20) Q q . Itfollows that for i ∈ { , } the mappings ~π i : B i → ~Q i given by ∀ k ∈ ω~π i ( X )( k ) = π i ( X ( k )) is a well defined continuous factor map. Consequently,they are Borel graph homomorphisms and since Q and Q are thin, we aredone.We can also use the colouring algorithm to deduce a property of thickgraphs that suggests the name is quite appropriate. First, note that beingthick means that the relation (cid:20) Q must be complex. Consequently, given abinary relation Φ ⊆ Q , it is possible for the BQO ≦ Q ∪ Φ to no longer bethick. For example, if Φ = Q . However, given the choice between Φ and itscompliment ¬ Φ, one of ≦ Q ∪ Φ or ≦ Q ∪ ( ¬ Φ) must be thick.
Proposition 1.
Let Q be thick. Let I = { Φ ⊆ Q : ≦ Q ∪ Φ thick } . Then I satisfies the following properties: • Ψ ⊆ Φ , Φ ∈ I ⇒ Ψ ∈ I . • Q / ∈ I . • ∅ ∈ I . • Φ / ∈ I ⇒ ¬ Φ ∈ I . This suggests there is a link between thick BQOs and directed sets (eg.filters and ideals). Q to Q <ω For this section, we will fix Q to be a BQO and R to be Q <ω under Higman’sorder. We also let B = { X ∈ ~R : ∀ k ∈ ω, len( X ( k )) ≤ len( X ( k + 1)) } . Definition 3.1.
Given X ∈ B , we let m k ( X ) ∈ ω be the largest integer suchthat X ( k ) ↾ m k ( X ) ≦ R X ( k + 1). We let n k ( X ) be the smallest integer suchthat X ( k ) ↾ m k ( X ) ≦ R X ( k + 1) ↾ n k ( X ).Note, for any X as in the above, m k ( X ) ≤ n k ( X ). Also, if X i ∈ B is asequence that converges to X , then ∀ k ∈ ω n k ( X i ) → n k ( X ) and m k ( X i ) → m k ( X ). Thus, both n k and m k are continuous for any k . They also satisfy n k ( S ( X )) = n k +1 ( X ) and m k ( S ( X )) = m k +1 ( X ).11 efinition 3.2. Given X ∈ B , if ∀ k ∈ ω , m k ( X ) ≤ m k +1 ( X ) < n k ( X ) ≤ n k +1 ( X ), then we define ∂X ∈ B by ∀ k ∈ ω , ∂X ( k ) = X ( k ) ↾ m k . This iswell defined since n k was minimal and m k +1 < n k . We call such X -derivable .We say ∂ X = X . Given i ∈ ω , we recursively define ∂ i X = ∂ ( ∂ i − X ) when ∂ i − X is 1-derivable. We call such X i -derivable
Derivations commute with S i.e ∂ i S ( X ) = S ( ∂ i X ). This is a consequenceof the previously noted fact n k ( S ( X )) = n k +1 ( X ) and m k ( S ( X )) = m k +1 ( X ).It follows from ∂ commuting with S that if X is i -derivable, then S ( X ) is i -derivable. Definition 3.3.
We let D ⊆ B be the set of all 1-derivable members from B . The mapping ∂ : D → ~R is a continuous factor map. Proposition 2. ∀ X ∈ B , there is a maximal i such that X is i -derivable.Proof. Suppose otherwise. Note len(( ∂ i X )(0)) = m ( ∂ i − X ) decreases inlength. We then have an infinite decreasing sequence in ω which is impossible. Definition 3.4.
For an X ∈ B , let M k ( X ) ∈ ω be the largest such ordi-nal such that S k ( X ) is M k ( X )-derivable. We define ∂ ∞ X by ∂ ∞ X ( k ) =( S k ( X )) ( M k ) (0).Note that for every k ∈ ω , the mappings M k ( X ) and ∂ M k ( X ) X are con-tinuous. Lemma 3.4.
The mapping X ∈ B defined by X → ∂ ∞ X is well defined andBaire class 1. Moreover, ∀ k ∈ ω S k ( ∂ ∞ X ) ∈ B \ D Proof.
First, since
X i -derivable ⇒ S ( X ) is i -derivable, M k ( X ) is an increas-ing sequence. Moreover, S k ( ∂ M k − ( X ) X ) is an initial segment of S k ( ∂ M k ( X ) X ).Consequently, ∂ ∞ X ( k ) = ( S ( k ) ( ∂ M k ( X ) X ))(0) (cid:20) R ( S ( k +1) ( ∂ M k ( X ) X ))(0) im-plies that ∂ ∞ X ( k ) (cid:20) R ∂ ∞ X ( k + 1) as ∂ ∞ X ( k + 1) is an initial segment of( S ( k +1) ( ∂ M k ( X ) X ))(0).To see the mapping X → ∂ ∞ X is Baire class 1, consider the mappings f j : ~R → ~R via ∀ i < j, f j ( X )( i ) = ( S i ( ∂ M i ( X ) X ))(0) f j ( X )( i ) = ( S i ( ∂ M j ( X ) X ))(0) otherwise12t is clear that these maps are continuous as finite derivations are continuous.It is also clear that f k ( X ) → ∂ ∞ X point-wise.Note that S ( ∂ ∞ X ) = ∂ ∞ S ( X ) as derivation commutes with S . The ex-istence of a k such that S k ( ∂ ∞ X ) ∈ D would contradict the maximality of M k ( X ) as we could have taken one further derivation. Lemma 3.5.
Given an X ∈ B , if m k +1 ( X ) ≥ n k ( X ) , the sequence Y ( k ) = X ( k )( m k ( X ) + 1) is in ~Q .Proof. Note, for all d ≥ n k ( X ), X ( k )( m k ( X ) + 1) (cid:20) Q X ( k + 1)( d ). Else, theexistence of such a d would mean X ( k ) ↾ m k ( X ) + 1 ≦ R X ( k + 1) ↾ d ≦ R X ( k + 1). Since m k ( X ) ≤ n k ( X ) ≤ m k +1 ( X ), ∀ k ∈ ω X ( m k ( X ) + 1) (cid:20) Q X ( m k +1 ( X ) + 1) as desired. Theorem 3.6. If Q is thin, then so is Q <ω .Proof. By our colouring algorithm, it suffices to Borel 3-colour the set B .To see this, consider the relation Φ( r, t ) ⇐⇒ len( r ) ≤ len( t ). Since thereis no infinite descending sequence in ω , we only need to worry about the Φhomogeneous members in ~R which is the set B . By lemma 3.4 and lemma2.1, the mapping ∂ ∞ : D → B \ D is a Borel graph homomorphism. Hence,it suffices to consider B \ D . Applying our algorithm twice, we can reducecolourability to the set of X ∈ B \ D that have the parameters m k ( X ) and n k ( X ) increasing. C = { X ∈ B \ D : ∀ k ∈ ω, m k ( X ) ≤ m k +1 ( X ) , n k ( X ) ≤ n k +1 ( X ) } Again using the algorithm, we can reduce to the case where ∀ k ∈ ω m k +1 ( X ) ≥ n k ( X ) or ∀ k ∈ ω , m k +1 ( X ) < n k ( X ). Note, the latter is impossible for mem-bers of C , else such an X would be in D . By lemma 3.5, { X ∈ C : m k +1 ( X ) ≥ n k ( X ) } (cid:22) B ~Q by the mapping f ( X )( k ) = X ( m k ( X ) + 1). Since Q is thin,we have Borel 3-colouring of ~R . Thus, Q thin implies Q <ω thin. Theorem 3.7. If Q is thin, then so is F T Q under ≦ .Proof. For each T ∈ F T , let ˆ T be a linear order extending the order of T .Given ( T, l ) ∈ F T Q , we can identify the pair ( ˆ T , l ) with a sequence in Q <ω .This is because of course, every finite linear order is uniquely determined bysize and a Q labeled linear order is then just a Q sequence. Notice that thismapping ( T, l ) → ( ˆ T , l ) satisfies (
S, m ) ≦ ( T, l ) ⇒ ( ˆ S, m ) ≦ Q <ω ( ˆ T , l ). Thisnaturally gives us a homomorphism from ~ F T Q into ~Q <ω via lemma 2.2.13 orollary 3.7.1. If Q is thin, so are [ Q ] <ω and F T Q under ≦ m .Proof. First, well order Q . Then, given an a ∈ [ Q ] <ω , assign to it the se-quence ˆ a ∈ Q <ω of its members placed in order according to the well order.Note that ˆ a ≦ Q ω ˆ b ⇒ a ≦ m b . Consequently, [ Q ] <ω is thin by theorem 3.6and lemma 2.2. Similarly for trees, ( S, m ) ≦ ( T, l ) ⇒ ( S, m ) ≦ m ( T, l ) for
A, B ∈ [ Q ] <ω and ( S, m ) , ( T, l ) ∈ F T Q . So, by lemma 2.2 and theorem 3.7 F T Q is thin under ≦ m . Since Laver showed that labeled σ -scattered orders are in some sense, nomore complicated than finite labeled trees, it comes as no surprise that if Q is thin, so are Q -labeled σ -scattered orders. We will prove this here. Westart by showing that any countable collection of Q labeled ordinals where Q is thin, is also thin under ≦ emb . Lemma 3.8.
Let Q be thin. Then for any cardinal κ , Q <κ is thin under ≦ emb Proof.
Let R be a countable subset of Q <κ . Applying our colouring algorithmto ~R , we need only consider A = { X ∈ ~R : len( X ( k )) ≤ len( X ( k + 1)) } Akin to in the finite case, for X ∈ A and k ∈ ω , there is an α k ( X ) , β k ( X ) < κ where α k ( X ) is minimal such that X ( k ) ↾ { β : β < α k ( X ) } (cid:20) emb X ( k + 1)and β k ( X ) is minimal such that X ( k ) ↾ α k ( X ) ≦ emb X ( k + 1) ↾ β k ( X ). Notewhen κ = ω , this is identical to the finite case. Since R is a countable subset,and members from R may be uncountable labeled ordinals, it may not bethe case that for every ( γ, l ) ∈ R and ζ < γ , ( ζ , l ↾ ζ ) ∈ R . However, we mayassume that R is closed under all relevant restrictions. First, we will provethis claim.Given a pair ( γ , l ) , ( γ , l ) ∈ Q <κ , we define α (( γ , l ) , ( γ , l )) to be theminimal α such that ( α, l ↾ α ) ≦ emb ( γ , l ) and β (( γ , l ) , ( γ , l )) the min-imal β such that ( α, l ↾ α ) ≦ emb ( β, l ↾ β ). Given R ⊆ Q <κ , we define the14ollowing sets R l = { ( α, l ) : ∃ ( γ , l ) , ( γ , l ) ∈ R, ( α, l ) = ( α (( γ , l ) , ( γ , l )) , l ↾ α (( γ , l ) , ( γ , l ))) } R r = { ( α, l ) : ∃ ( γ , l ) , ( γ , l ) ∈ R, ( α, l ) = ( α (( γ , l ) , ( γ , l )) , l ↾ α (( γ , l ) , ( γ , l ))) } R ′ = R l ∪ R r Note that if R is countable, so is R ′ as α ( · , · ) is parameterized over pairsfrom R of which there are only countably many. Given an R , we definerecursively ∀ i ∈ ω , R = R , R i = ( R i − ) ′ . The set R fill = ω S i =0 R i con-tains R is a subset and has the property that for any ( γ , l ) , ( γ , l ) ∈ R fill ,( α (( γ , l ) , ( γ , l )) , l ↾ α (( γ , l ) and ( α (( γ , l ) , ( γ , l )) , l ↾ α (( γ , l ) are in R fill . Consequently, we could define a derivative operation as we did in thefinite case on ~R fill ⊇ ~R . We will explain why this is the case next, but fornow we’ve at least shown that we can assume R is sufficiently closed underrestrictions.Note that it is never the case that α k ( X ) = dom( X ( k )). If it were, then foreach β ∈ dom( X ( k )), there is an h ( β ) such that X ( k )( β ) ≦ Q X ( k + 1)( h ( β ))and h strictly increasing. But of course, this h codifies that X ( k ) ≦ R X ( k +1), a contradiction. It is also the case that ∀ γ ≥ β k ( X ), X ( k )( α k ( X )) (cid:20) Q X ( k +1)( γ ) akin to the finite case. This means that the sequence X ( k )( α k ( X )) ∈ ~Q when α k ( X ) ≤ β k ( X ) < α k +1 ( X ).Similar to before, we can say an X is derivable if ∀ k ∈ ω , α k ( X ) ≤ α k +1 ( X ) ≤ β k ( X ) ≤ β k +1 ( X ). We can also define the derivation operation ∂ and it’s iter-ates by restricting to α k ( X ) coordinate-wise as before. Since we can assume R is closed under these coordinate-wise restrictions, the derivative operationis well defined on ~R . Moreover, ∂ is a Borel graph homomorphisms, andevery X is at most i derivable for some i as there is no infinite descendingsequence of ordinals. Consequently, we can define the operator ∂ ∞ for deriv-able members of ~R . This mapping is also well defined. It is also a Borelfactor map and mimicking the argument from the finite case, we have that ~R is thin. Theorem 3.9. If Q is thin, then so is T Q under ≦ .Proof. Let R be a countable subset of T Q . Let I = { T : ( T, l ) ∈ R } . Notethat each tree T ∈ I can have its order extended to a well order. Simply15ell order each set of immediate successors from T and then give T thelexicographical order. Given a T ∈ I , we call the well order ˆ T . Since I is countable, there is a cardinal κ such that each ( ˆ T , l ) ∈ Q <κ . Since theordering extends the tree order, we have ( T , l ) (cid:20) R ( T , l ) ⇒ ( ˆ T , l ) (cid:20) Q <κ ( ˆ T , l ). It follows that ~R is thin by our last lemma. Hence, T Q is thin. Corollary 3.9.1. If Q is thin, then the class of Q labeled σ -scattered linearorders is thin.Proof. It suffices to show Q + is thin. Since the disjoint union of thin sets arethin, it suffices to show that RC and RC × RC are thin. Since products ofthin sets are thin, we need only show that RC is thin. Given any R ⊆ RC countable, ~R is empty as there is no infinite descending sequence of ordinals.Hence, RC is thin as desired. The author would like to thank Yann Pequignot for their email correspon-dence, as well as for sharing some notes on the problem. The author wouldalso like to thank the Calgary discrete mathematics seminar for allowing theauthor to present his initial findings. Finally, the author would like to thankStevo Todorcevic for introducing him to the problem, and supporting himthrough the solution process. 16 eferences [1] R. Fra¨ıss´e, Sur la comparaison des types d’ordres.
C. R. Acad. Sci.
Paris226 (1948)[2] F. Galvin and K. Prikry. Borel sets and Ramsey’s theorem.
The Journalof Symbolic Logic.
Math.Ann.
65 (1908), 435-505.[4] G. Higman, Ordering by divisibility in abstract algebras,
Proc. of LondonMathematical Society.
GraduateTexts in Mathematics.
Springer-Verlag, New York, 1995.[6] A. S. Kechris, A. Louveau. A Classification of Baire Class 1 Functions.
Tansactions of the American Mathematical Society.
Vol. 318 No. 1 pp209-236. 1990[7] A. S. Kechris and A. S. Marks. Descriptive graph combinatorics. 2015.
Available at http://math.ucla.edu/ marks/[8] A. S. Kechris, S. Solecki, and S. Todorcevic. Borel chromatic numbers.
Adv. Math. , 141(1):1-44, 1999.[9] Kenneth Kunen, Set Theory.
College Publications.
The Annals ofMathematics.
93 (1): 89-111.[11] Richard Laver, An Order Type Decomposition Theorem.
Annals ofMathematics , Second Series, Vol. 98, No. 1 (1973), pp. 96-119[12] A. Marcone. Foundations of bqo theory.
Trans. of the American Math-ematical Society. -complete. Mathe-matical Logic Quarterly.
Available at
Math-ematical Proceedings of the Cambridge Philosophical Society.
61 (3): 697-720, 1965[16] C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequence.
Proc. Cambridge Phil. Soc.
61 (1965), 33-39.[17] Y. Pequignot. Finite versus infinite: an insufficient shift.
Adv. Math. ,320(1): 244-249, 2017[18] M. Pouzet. Applications of Well Quasi-Ordering and Better Quasi-Ordering.
Graphs and Order. vol. 147. pp 503-519, 1984[19] S. Todorcevic and Z. Vidnyanszky. A complexity problem for Borelgraphs. submitted,submitted,