A collection of topological Ramsey spaces of trees and their application to profinite graph theory

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1 A collection of topological Ramsey spaces of trees and their application to profinite graph theory Yuan Yuan Zheng Abstract We construct a collection of new topological Ramsey spaces of trees. It is based on the Halpern-Läuchli theorem, but different from the Milliken space of strong subtrees. We give an example of its application by proving a partition theorem for profinite graphs. 0 Introduction Milliken s theorem ([8] concerns colourings of all infinite strong subtrees of a fixed tree U, where U is an arbitrary rooted finitely branching tree of height ω without terminal nodes. It shows that the Ramsey property for sets of infinite strong subtrees is captured by the Baire property in the corresponding tree topology. In other words, as Todorcevic [10] formulates it, the set of all infinite strong subtrees of U form a topological Ramsey space, called the Milliken space. The proof of Milliken s theorem is based on the Halpern-Läuchli theorem ([5]. Milliken s theorem generalizes Silver s theorem ([7] and has found various applications to partition properties. For example, Devlin in [2] defined the Devlin embedding types of finite subsets of the complete binary tree, which can be realized by going to a strong subtree, and applied Milliken s theorem to show that the odd tangent numbers characterize the Ramsey properties of the countable dense linear ordering (Q, < Q. Moreover, most of the known perfectset theorems can naturally be proved on the basis of the Milliken space. For example, the following partition theorem for perfect sets was conjectured by Galvin ([3], who proved it for n 3, and proved by Blass. Theorem 0.1 (Blass. [1]. For every perfect subset P of R and every finite continuous colouring of [P ] n there is a perfect set Q P such that [Q] n has at most (n 1! colours. To prove this theorem, Blass defined patterns, or embedding types, for finite subsets of a perfect tree T such that for every finite continuous colouring of finite subsets of the nodes in T, one can make all subsets with a fixed pattern monochromatic by going to a perfect subtree. Todorcevic ([10, Cor. 6.47] provided a simpler proof of Blass s theorem using the Milliken space, as the perfect trees in Blass s argument can be replaced by strong subtrees Mathematics Subject Classification: 03E02, 03E05, 05C05, 05C55. Key words: Ramsey theory, topological Ramsey space, trees, partition, profinite graph theory 1

2 In this paper, we construct a collection of topological Ramsey spaces of trees, one for each type τ of finite graphs. Each space (G (τ,, r consists of G max - trees of a particular shape. Similar to the Milliken space, the new spaces G (τ are also based on the Halpern-Läuchli theorem. We expect the spaces to have similar applications as the Milliken space does. We exemplify this by applying these topological Ramsey spaces to prove the following partition theorem for profinite graphs. Theorem 0.2 (Huber-Geschke-Kojman. For every finite ordered graph A there is k(a < ω such that for every universal modular profinite ordered graph G, and for every finite Baire-measurable colouring of the set of all copies of A in G, there is a closed copy G G of G such that the set of all copies of A in G has at most k(a colours. This theorem was originally proved by Huber, Geschke, and Kojman in [6]. Similar to the way Blass proved Theorem 0.1 by partitioning finite subsets into embedding types in [1], Huber, Geschke, and Kojman proved the above theorem by partitioning the isomorphism class of A into k(a many smaller classes, called strong isomorphism classes, or types. We use the new space G (τ to prove the following Theorem 0.3 from which Theorem 0.2 follows as shown in [6]. Theorem 0.3. [6, Thm. 3.2]. Let T be an arbitrary G max -tree. For every type τ of a finite induced subgraph of G max, and for every continuous colouring c : ( G(T τ 2, there is a Gmax -subtree S of T such that c is constant on ( G(S τ. In the same manner as Todorcevic proved Blass s Theorem 0.1, we design a G -tree envelope for every finite ordered graph, and use the properties of the topological Ramsey space G (τ to prove Theorem 0.3. We define G max -trees and types in Section 1, construct the topological Ramsey spaces in Section 2, and prove the partition theorem for profinite ordered graphs as an application of the spaces in Section 3. I am grateful to Professor Stevo Todorcevic for his guidance. I would also like to thank Osvaldo Guzmán González who pointed out a mistake in an earlier version of this work. This work was partially supported by the Queen Elizabeth II Graduate Scholarship in Science and Technology. 1 Preliminaries 1.1 Trees and graphs By a graph we mean an ordered graph. When we say that two graphs are isomorphic, we tacitly understand that they are order-isomorphic. Similarly, by an embedding we mean an order-embedding. For a set X, we say Y is an n-subset of X if Y is a subset of X of size n. Let [X] n = {Y X : Y = n} be the set of all n-subsets of X. For a graph G, let V (G denote its vertex set and E(G [V (G] 2 denote its edge relation. Definition 1.1. Let ω <ω be the set of all finite sequences of elements in ω. Let denote the initial segment relation. For an element s ω <ω, let s denote the length of s. For n < ω, let ω <n = {s ω <ω : s < n} and ω >n = {s ω <ω : s > n}. 2

3 The definitions of ω n and ω n are similar. By a tree we mean a downward closed subset T of ω <ω, ordered by. Each element t of a tree T is called a node. For a tree T, let [T ] be the set of all infinite branches of T, i.e. [T ] = {x ω ω : ( n < ω x n T }, where we consider x as an infinite sequence of natural numbers, and x n is its initial segment of length n. We say T is a subtree of T if T T and T is a tree. Definition 1.2. For a tree T and t T, let succ T (t be the set of all immediate successors of t in T, i.e. succ T (t = {s T : (t s ( s T (s s s t}. Let T t be the set of all nodes in T comparable to t, i.e. T t = {s T : t s s t}. Now we define the tree T max, which is the base of the construction of the topological Ramsey space G. Let R be the Rado graph, with vertex set ω. For n ω, let R n be the induced subgraph of R on {0,..., n}. Definition 1.3. [6]. Let T max ω <ω be the tree such that t T max succ Tmax (t = {t 0, t 1,..., t t }. For t T max, we define G t to be the ordered graph on the vertex set succ Tmax (t with lexicographical ordering, such that G t is isomorphic to R t. Note that [T max ] ω ω. Suppose x, y [T max ], let x y ω <ω be the common initial segment of x and y, i.e. x y = x min{n : x(n y(n}. Definition 1.4. [6]. The tree T max and the graphs G t (t T induce a graph G max on the vertex set [T max ], ordered lexicographically, with the edge relation defined as follows. For x, y [T max ], {x, y} E(G max if and only if {x ( x y + 1, y ( x y + 1} E(G x y. Lemma 1.5. [6, Lem. 2.1]. If H is a finite graph and t T max then there is s T max with t s such that H embeds into G s. We construct large subtrees of T max, the G max -trees, to get induced subgraphs of G max containing copies of G max. Definition 1.6. [6]. Let T be a (finite or infinite subtree of T max and t T. We define G T t to be the induced subgraph of G t on the vertex set succ T (t. We define G(T to be the induced subgraph of G max on [T ]. We say T is a G max -tree if for every finite graph H and every t T, there is s T with t s such that H embeds into G T s. In particular, T max is a G max -tree. 3

4 For n ω, let [G max ] n denote the set of all induced subgraphs of G max of size n. Since each such subgraph is completely determined by its vertex set, we may identify [G max ] n with the set of all n-subsets of vertices of G max, i.e. [G max ] n = [V (Gmax ] n = [[T max ]] n [ω ω ] n. We equip ω ω with the first-difference metric topology, which has basic open sets of the form [σ] = {x ω ω : σ x} for σ ω <ω. Then we let [ω ω ] n have the induced product topology, and [G max ] n have the subspace topology. 1.2 Types Definition 1.7. Suppose T is a G max -tree and H is a finite induced subgraph of G(T. Let (H = max{ x y : x, y H}, H = {x ( (H + 1 : x H}. We illustrate the idea of this definition with an example. Example. Let H = {x, y, z} [G max ] 3 as in Figure 1, where x = , y = , and z = Then (H = 2 and H = {s, t, u}, where s = 000, t = 002 and u = 011. x y z... s t u Figure 1: (H and H Definition 1.8. [6]. Let T be a G max -tree. Let H and H be finite induced ordered subgraphs of G(T. We say H and H are strongly isomorphic if there exists an isomorphism ϕ : H H such that {x 0, y 0 }, {x 1, y 1 } [H] 2, x 0 y 0 x 1 y 1 ϕ(x 0 ϕ(y 0 ϕ(x 1 ϕ(y 1. Clearly, strong isomorphism is an equivalence relation. By a type we mean a strong isomorphism equivalence class. We have the following observation: Let H be a finite induced subgraph of G(T, with T being a G max -tree. We order its vertex set V (H = {x 0,..., x k 1 } lexicographically. We consider the way in which x i x i+1 (i < k 1 is ordered in the usual ordering of ω. 4

5 If two graphs H, H are order-isomorphic and have the same ordering of their corresponding x i x i+1 (i < k 1, then they are strongly isomorphic. In particular, there are only finitely many types inside an isomorphism class. Definition 1.9. Suppose G, H are graphs and τ is a type. Let ( G H be the set of all induced subgraphs H of G isomorphic to H. Let ( G τ be the set of all induced subgraphs of G of type τ. 2 A topological Ramsey space In this section, we fix a type τ and build a topological Ramsey space. We may denote this space by G (τ to emphasize the type. We write it as G when the type is clear from the context. Let m + 1 be the number of vertices for a graph in τ. Let {F i : i < ω} be the set of all finite (ordered graphs, up to isomorphism, labelled such that for every i < j < ω, either V (F i < V (F j or V (F i = V (F j and E(F i E(F j. Let T be a tree. We say a node t T is a branching node if succ T (t > 1. We say T is skew if T has at most one branching node at each level, i.e. n ω {t T ω n : succ T (t > 1} 1. Recall that for a (finite or infinite tree T and a node t T, T t = {s T : t s s t}. Definition 2.1. We define the space (G,, r as follows. Let A G if A is a skew subtree of T max and when we enumerate the set of branching nodes {s A : succ A (s > 1} as {s i } i<ω in the order of length, (1 there is a graph H τ with (2 for all i > 0, H = A ω sm +1 ; (i s A ω > sm u succ A (s G A s = F i!t A ( u t G A t = F i+1 ; (ii for every pair t, s A ω > sm, G A t = F i G A s = F i+1 t < s ; (iii if {t A : G A t = F i } is enumerated in the order of length t, as {t j } j, then there is k < ω such that {t j k} j is strictly increasing in lexicographical ordering. When we say that {s i } i<ω is the set of branching nodes in A, we tacitly assume that the length s i is strictly increasing in i. For A, B G, let A B if A B. 5

6 For k < ω and A G with the set of branching nodes {s i } i<ω, we define the finite approximation r k (A inductively on k. Let r 0 (A =. Let r k+1 (A = A ω sk +1. Intuitively, (1 requires A to have a base of the same shape as H, for some H τ. Condition (2 indicates that we can divide the rows above the base of A into interval sections such that in the ith section we see that each node at the bottom of the section extends and branches into precisely one fan whose induced subgraph is isomorphic to F i. Moreover, the branching nodes are taller and taller when we look from left to right. An example of an element A G and some of its finite approximations are shown in Figures 2 to 5.. s 8 s 7.. s 5 s 6 G A s i = F2 s 4 s 3 G A s i = F1 s 2 s 1 base s 0 Figure 2: Example of A G s 0 Figure 3: r 1 (A s 1 s 0 Figure 4: r 2 (A s 2 s 3 s 1 s 0 Figure 5: r 4 (A Before checking that (G,, r is a topological Ramsey space, we specify a few more definitions that are often used in topological Ramsey spaces. Let G < denote the set of all finite approximations, i.e. G < = {r k (A : A G k ω}. 6

7 For a, b G <, let a fin b if a b. Let a = n if there is A G with r n (A = a. For a, b G <, we write a b if there are k < l < ω and A G such that a = r k (A and b = r l (A. Note that the same symbols and are used in two different situations: among elements in ω ω, and among elements in G <. For a G < and A G, depth A (a = min{n : a fin r n (A}, where by convention min =. We equip the space G with the Ellentuck topology, with basic open sets of the form [a, A] = {X G : (X A ( k(r k (X = a}, for a G < and A G. For k < ω, let G k = {r k (X : X G } [k, A] = [r k (A, A], and r k [a, A] = {r k (X : X [a, A]}. The height height(a of an element a G < is n if s = n for all s a. In general, height(a a. Now we show that (G,, r is a topological Ramsey spaces by proving that G is closed and it satisfies the axioms (A1-(A4 as defined in [10]. It is straightforward to check (A1-(A3 and that G is a closed subset of (G < ω when we identify A G with (r n (A n<ω (G < ω and equip G < with the discrete topology and (G < ω the product topology. We prove (A4 in Lemma 2.5 below. We will be using the Nešetřil-Rödl Theorem 2.2 and the Halpern-Läuchli Theorem 2.4. Theorem 2.2 (Nešetřil-Rödl. For every finite ordered graphs A and B there is a finite ordered graph C such that for every 2-colouring of ( C A there is B ( C B such that ( B A is monochromatic. In this case, we write C (B A. In fact, we only need the 1-dimension version of the Halpern-Läuchli theorem below. Definition 2.3. Let T be a (downwards closed tree and N an infinite subset of ω. We say a set S is a strong subtree of n N T ωn if there is an infinite set M N such that 1. S m M T ωm and S ω m for all m M; 2. if m 1 < m 2 are two successive elements of M and if s S ω m1, then every immediate successor of s in n N T ωn (ordered by has exactly one extension in S ω m2. Note that a strong subtree is not necessarily a tree according to our Definition 1.1 since it may not be downward closed. Theorem 2.4 (Halpern-Läuchli. For a tree T and N [ω] ω, for every finite colouring of n N T ωn there is a strong subtree S of n N T ωn with M [N] ω such that m M S ωm is monochromatic. 7

8 We observe that every A G is a G max -tree and every G max -tree contains some A G as a subtree. This is because for all n < ω, we can find infinitely many m ω such that F n embeds into F m. We define (J n n<ω to be a sequence of finite graphs such that for each n < ω the graphs F 0,..., F n embed into J n. Lemma 2.5. The axiom (A4 holds for (G,, r, i.e. B G, if depth B (a < and O G a +1, then for a G < and A [depth B (a, B] r a +1 [a, A] O or r a +1 [a, A] O =. Proof. Let a, B, O be as in the statement. Let h 0 = height(a. Note that a has a many branching nodes, and each element in G a +1 has a + 1 many. In particular, we can find u a ω h0 and i 0 < ω such that for every b G a +1 with a b there is a unique branching node t b u, and G b t =. We define a subset B { ( G B } t : t Bu induced by O as follows. For t B u and H ( G B t, let b be the downward closure of a V (H. Put H in B if and only if b O. Let B c denote { ( G B } t : t Bu \ B. We aim to shrink the tree B u to a G max -tree A such that { ( G A t : t A B or B c. Then it is clear that for each v a ω h0 \ {u} we can shrink B v to a tree A v such that A A v G. Thus, letting A = A A v we have A [a, B] and r a +1 [a, A] O or r a +1 [a, A] O = as required. We first shrink B u to X, and then X to A. We construct X with a strictly increasing sequence (m k k<ω, where m 0 = 0, such that k < ω s X ω m k, (a X s has a unique branching node t of length in [m k, m k+1 ; (b moreover, for the t from condition (a, G X t ( (c t X G X t s B or B c. = J i0+k; We recursively construct sets X(m k ω m k, then X will be the downward closure of k<ω X(m k and X ω m k = X(m k. Start with m 0 = 0 and X(m 0 = { }. Suppose we have constructed X(m k = {s j : j < l}. By Theorem 2.2, there is a finite graph H such that H (J i0+k Fi 0. For s j X(m k, since B is a G max -tree, s j has an extension t j B such that H embeds into G B t j. By the choice of H, we can find K j ( G B t j J such that i0 +k ( Kj B or B c. Let m k+1 = max{ t j : j < l} + 1. Let X(m k+1 B ω m k+1 such that every node in j<l V (K j has a unique extension in X(m k+1. This finishes the construction of X and (m k k<ω. Now we define a colouring c : k<ω X ωm k 2. For s X ω m k, let c(s = 1 if and only if the unique branching node t X s of length in [m k, m k+1 from (a satisfies ( G X t B. By Theorem 2.4, there is a strong subtree Y of k<ω X ωm k and a strictly increasing sequence (n j j<ω (m k k<ω such that c j<ω Y ωnj ɛ for some ɛ 2. Let A be the downward closure of Y. } 8

9 Claim If k < ω, s Y ω m k and t corresponds to s as in (a, (b, then t A and G A t = G X t. Proof of Claim. As Y ω mk, there is l < ω such that n l = m k. Since Y is a strong subtree of k <ω X ωm k, every immediate successor of s in k <ω X ωm k (that is, the elements in X ω m k+1 extending s has exactly one extension in Y ω n l+1. From the construction of X we know that every node in succ X (t has exactly one extension in X ω m k+1. But the successors of s in k <ω X ωk are precisely the extensions in X ω m k+1 of nodes in succ X (t. Hence the downward closure A of Y must contain t as an element and satisfy G A t = G X t. Then it is straightforward to check that A is a G max -tree. We check that ( G A t A t B or B c. Suppose t A is a branching node. Since A X, t must also be a branching node in X. We can find k such that t [m k, m k+1. Then, by the construction of A, as Y is a strong subtree of X, there is l such that m k = n l, so Y ω n l X ω m k. Let s Y ω n l be an initial segment of t. As c j<ω Y ωnj ɛ, c(s = ɛ. Therefore ( G A t B or B c, depending on the value of ɛ. This holds for all branching nodes t A, so ( G A t A t B or B c as required. This finishes the proof of the lemma. Theorem 2.6. The space (G,, r is a topological Ramsey space. The corollary below follows from the abstract Ellentuck theorem [10, Thm. 5.4]. Corollary 2.7. For every finite Souslin-measurable colouring of G and every A G there is B A such that [, B] is monochromatic. 3 An application to profinite graph theory In this section we present an application of the topological Ramsey spaces (G (τ,, r to profinite graph theory. We define a G -envelope for each finite induced subgraph H. We then prove Theorem 0.3. A modular profinite ordered graph is the inverse limit of an inverse system of finite ordered graphs with modular bonding maps. (See [4] for details. A modular profinite ordered graph is universal if every modular profinite ordered graph order-embeds continuously into it. Geschke proved the existence of a universal modular profinite ordered graph in [4]. Below is a definition of universal profinite ordered graphs from [6] without mentioning an inverse system. Definition 3.1. [6]. A universal profinite ordered graph is a triple G = V, E, <, such that the following conditions hold. 1. V is a compact subset of R \ Q without isolated points, E [V ] 2, and < is the restriction of the standard order on R to V. 2. For every pair of distinct vertices u, v V there is a partition of V to finitely many closed intervals such that (a u, v belong to different intervals from the partition; 9

10 (b for every interval I in the partition, for all v V \ I and for all x, y I, {v, x} E if and only if {v, y} E. 3. Every nonempty open interval of V contains induced copies of all finite ordered graphs. It is clear from this definition that G(T is a universal profinite ordered graph for every G max -tree T. It thus follows from the universality that in order to prove Theorem 0.2, it is sufficient to consider colourings of finite induced subgraphs of G(T, where T is an arbitrary G max -tree. As the number k(a of types in the isomorphism class of a finite ordered graph A is finite, the problem reduces to finding a monochromatic set for colourings of finite induced subgraphs of a fixed type. So it is sufficient to prove Theorem 0.3, which extends readily to Bairemeasurable colourings using standard methods from descriptive set theory. See [6] for a detailed description of how Theorem 0.2 reduces to Theorem 0.3. Definition 3.2. Let τ be a type and H τ. The G -envelope of H is C H = {A G (τ : ( k(r k (A = H }. Lemma 3.3. Let τ be a type and T G (τ. The map { ( } G(T c 1 : [, T ] H : H τ A H if A C H is well-defined and continuous, where we equip the range with discrete topology. Proof. First we check that c 1 is well-defined. Let m < ω be such that, for every H τ, H has m + 1 branching nodes. Then A G k (r k (A = H k = m + 1. Thus, if H, K τ and A C H C K then H = r m+1 (A = K. So c 1 is well-defined. Next, we check that c 1 is continuous. Suppose H τ and A (c 1 1 (H, i.e. A C H. Then the set [m + 1, A] is an open neighbourhood of A included in the c 1 -preimage of H. Hence c 1 is continuous. Finally we prove Theorem 0.3. We will use the following lemma. Lemma 3.4. [6, Lem. 3.3] Let T G and τ be a type. For every continuous colouring c : ( G max τ 2 there exists S T such that c depends only on H, i.e. for H, K ( G(S τ, if H = K then c(h = c(k. Theorem 0.3. [6, Thm. 3.2]. Let T be an arbitrary G max -tree. For every type τ of a finite induced subgraph of G max, and for every continuous colouring c : ( G(T τ 2, there is a Gmax -subtree S of T such that c is constant on ( G(S τ. Proof. Let c and T be given as in the theorem. By Lemma 3.4, we may shrink T and assume that the colouring c depends only on H. We further shrink T and assume T G. We may think of c as a function { ( } G(T c : H : H 2. τ 10

11 Define c : [, T ] 2 by letting c = c c 1. By Lemma 3.4, c is also a continuous function. In particular, c is Souslin-measurable. Now applying Corollary 2.7 gives S T and i 2 such that c [, S] i. Let us check that S satisfies the theorem. Suppose H ( G(S τ. Then its vertex set V (H [S], so H S. Hence there is A [, S] such that A C H. So c(h = c (A = i. Therefore c ( G(S τ i as required. References [1] A. Blass. A partition theorem for perfect sets. Proc. Amer. Math. Soc. 82 (1981, no. 2, [2] D. Devlin. Some partition theorems and ultrafilters on ω. Ph.D. Thesis, Dartmouth College, [3] F. Galvin. Partition theorems for the real line. Notices Amer. Math. Soc. 15 (1968, 660. [4] S. Geschke. Clopen graphs. Fundamenta Mathematicae 220, No. 2 (2013, [5] J. Halpern and H. Läuchli. A partition theorem. Trans. Amer. Math. Soc. 124 (1966, [6] Stefanie Huber, Stefan Geschke, and Menachem Kojman. An induced Ramsey theorem for modular profinite graphs. uni-hamburg.de/home/geschke/papers/gmaxsubjuly28.pdf [7] Silver, Jack. Every analytic set is Ramsey. J. Symbolic Logic 35 (1970, [8] K. R. Milliken. A partition theorem for the infinite subtrees of a tree. Trans. Amer. Math. Soc. 263 (1981, no. 1, [9] J. Nešetřil, V. Rödl. A structural generalization of the Ramsey theorem. Bull. Amer. Math. Soc. 83 (1977, no. 1, [10] S. Todorcevic. Introduction to Ramsey spaces, volume 174 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ,