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Age and weak indivisibility
European Journal of Combinatorics 37 (2014) 24–31
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European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc
Age and weak indivisibility N. Sauer University of Calgary, Department of Mathematics and Statistics, University Drive 2400, Calgary, T2N 1N4, Alberta, Canada Institute of Discrete Mathematics and Geometry, University of Technology in Wien (Vienna), Austria
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Article history: Available online 9 August 2013
abstract A relational structure H is homogeneous if every isomorphism between finite induced substructures has an extension to an automorphism of H. A relational structure R is indivisible if for every partition (A, B) of the set of elements of R, there is a copy of R in A or in B. The structure R is weakly indivisible if for every partition (A, B) of the set of elements of R for which some finite induced substructure of R does not have a copy in A, there exists a copy of R in B. The structure R is age indivisible if for every partition (A, B) of the elements of R every finite induced substructure of R has an embedding into A or every finite induced substructure of R has an embedding into B. It follows that indivisibility implies weak indivisibility which in turn implies age indivisibility. There are many examples of countable infinite homogeneous structures which are weakly indivisible but not indivisible; see Sauer (2000) [17] and the end of Section 3.1 of this paper. In general, it is difficult to find structures which are age indivisible but not weakly indivisible. One such example (see Pouzet et al. (2011) [16]) is obtained via the inhomogeneous general linear group of vector spaces over finite fields. Finding such an example for homogeneous structures seems to be even more difficult. The only one that I obtained, together with L. Nguyen Van Thé, is a countable homogeneous metric subspace of the Hilbert sphere ℓ2 (see Nguyen Van Thé and Sauer (2010) [13]). Both of the examples above are relational structures with infinitely many relations. One of the still open questions then is: Are there countable infinite homogeneous structures with finite signature which are age but not weakly indivisible? We will show that the generic ‘‘free amalgamation’’ homogeneous structures are weakly indivisible; see Section 3.1. Countable ‘‘Urysohn metric spaces’’ (see the next section), with finite distance sets, are indivisible (see Sauer (2013) [19]).
E-mail address: [email protected]. 0195-6698/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ejc.2013.07.013
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Countable Urysohn metric spaces are age indivisible, which follows from the Hales–Jewett theorem (Hales and Jewett (1963) [7]; see Delhommé et al. (2007) [2]). For non-Urysohn homogeneous countable metric spaces, except for the example mentioned above, the weak indivisibility question is completely open. We will show, in this paper, that a countable homogeneous ultrametric space is age indivisible if and only if it is weakly indivisible. © 2013 Elsevier Ltd. All rights reserved.
1. Basic notions See [6,17] for a more complete introduction to homogeneous structures and the theory of partitions of such structures. Relational structures are denoted with capital Roman letters and the same letter in slanted format (italics) is used to indicate the corresponding base set. Let R be a relational structure and S ⊆ R. Then R ↓ S denotes the restriction of R to S, that is the substructure of R induced by S. A copy of the relational structure R in the relational structure B is an induced substructure of B isomorphic to R. The relational structure R can be embedded into the structure B if it has a copy in B. The age of R consists of all finite structures which can be embedded into R. A countable set of finite relational structures having the same signature is an age if it is closed under isomorphism and induced substructures and is updirected, that is for any two structures in the age there is one into which both can be embedded. An age is infinite if it has infinitely many isomorphism types and countable if it has countably many isomorphism types. For every countable age A there is a countable structure R whose age is A; see [6]. An age A is a vertex Ramsey class if for every structure R ∈ A there is a structure S ∈ A such that for every partition (A, B) of S there is a copy of R in A or in B. A local isomorphism of A is an isomorphism of a finite induced substructure of A to an induced substructure of A. A relational structure H is homogeneous if every local isomorphism of H has an extension to an automorphism of H. Let R be a countable relational structure; then R is: 1. Age indivisible if for every partition (A, B) of the elements of R every finite induced substructure of R has an embedding into A or every finite induced substructure of R has an embedding into B. 2. Weakly indivisible if for every partition (A, B) of R for which there exists an element A in the age of R which cannot be embedded into R ↓ A, the structure R can be embedded into R ↓ B. 3. Indivisible if for every partition (A, B) of R there is a copy of R in A or in B. A countable relational structure R is age indivisible if and only if its age is a vertex Ramsey class; for a proof see [5]. For some earlier results on indivisible ages and indivisible relations by M. Pouzet, see [14,15]. There are two large classes of prominent countable infinite homogeneous structures. One of them is the class of homogeneous structures whose age has free amalgamation; see Section 3.1. According to Theorem 3.2 all of those homogeneous structures are weakly indivisible. For most signatures a complete classification of homogeneous structures whose age does not have free amalgamation is completely beyond our abilities. There are some sporadic cases such as the countable infinite graph whose every connected component is a triangle, or more generally the complete graph on n vertices for a fixed n. Of course in such cases the divisibility problems can easily be resolved. On the other hand to resolve the partition question for the dense local order has not been easy; see [11]. For the case of vector spaces some special cases are discussed in [16]. The other one is the class of countable homogeneous metric spaces. An Urysohn metric space with V as set of distances is a homogeneous complete metric space which embeds every separable, and hence every finite, metric space F whose set of distances is a subset of V , the classical Urysohn metric space being the unique complete separable metric space which embeds every separable metric space—in this case V being the set of nonnegative reals. Countable Urysohn metric spaces are age indivisible;
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see [2]. Unfortunately we cannot decide whether such spaces are weakly indivisible, not even for the most prominent one, the rational Urysohn sphere, that is for the countable Urysohn space whose set of distances is the interval [0, 1] ∩ Q. We can only prove that it is ‘‘approximately weakly indivisible’’; see [13]. Countable Urysohn metric spaces with a finite set of distances are indivisible (see [19]) and the countable Urysohn metric space with distance set the nonnegative integers is weakly indivisible (see [12]). Corollary 2.1 settles the case of countable ultrametric spaces: a denumerable homogeneous ultrametric space is weakly indivisible if and only if it is age indivisible. 2. Ultrahomogeneous metric spaces Let f be a function of a set S to ω. Then supp(f ) := {s ∈ S : f (s) ̸= 0}. Let λ be a countable chain, that is a linearly ordered countable set. A function a : λ → ω with 2 ≤ a(µ) ≤ ω for all µ ∈ λ will be called a degree function for λ. Let a be a degree function for λ; then
ω[a] := {f : λ → ω | ∀µ ∈ λ (f (µ) < a(µ)) and supp(f ) is finite}. So for example let λ = ω and a(n) = 2 for all n ∈ ω. Then every f ∈ ω[a] is a function from ω into {0, 1} := 2 for which there are only finitely many n ∈ ω with f (n) = 1 and with f (n) = 0 or f (n) = 1 for all n ∈ ω. It follows that ω[a] is a subtree of the Cantor tree, that is the tree of height ω whose branches are the set of all 0, 1-sequences. In this case, for the distance ∆ defined below, we have that ∆(f , g ) is the smallest level of the tree, that is the smallest n ∈ ω, for which f (n) ̸= g (n). Associate with every level of the tree, that is with every positive integer n, the number 1n . Let d(f , g ) := 1n with n = ∆(f , g ). Then it turns out (see Theorem 2.1 below) that ω[a] in this case is an ultrametric homogeneous space with distance function d. If a(µ) = ω for every µ ∈ λ, then the set ω[a] is usually denoted by ω[λ] . For f , g ∈ ω[a] let
∆(f , g ) :=
∞, µ,
if f = g ; for µ the smallest element in λ with f (µ) ̸= g (µ).
For M a metric space and a ∈ M let Spec(M, a) := {d(a, x) | x ∈ M } and Spec(M) := {d(x, y) | x, y ∈ M } and Ba (s) := {x ∈ M | d(a, x) ≤ s} and Nerv(M ) := {Ba (s) : a ∈ M , s ∈ Spec(M , a)}. An ultrametric space M is a metric space with d(x, y) ≤ max{d(x, z ), d(y, z )} for all x, y, z ∈ M. The following theorems, Theorems 2.1 and 2.2, are from [2]; see [3] as well. Theorem 2.1. Let λ be a countable chain and a a degree function for λ and w : λ ∪ {∞} → R+ a strictly decreasing map with w(∞) = 0 and image V . Let dw := w ◦ ∆. Then M = (ω[a] , dw ) is an ultrametric space with Spec(M) = V . Let M be a countable ultrametric space. Then the following properties are equivalent: a. M is isometric to some (ω[a] , dw ). b. M is homogeneous. c. M is point-homogeneous. (That is, the automorphism group Iso(M) of M acts transitively on M.) The space (ω[λ] , dw ) is the countable homogeneous ultrametric space UltV associated with V . Theorem 2.2. Let M be a denumerable ultrametric space. Then the following properties are equivalent: i. M is isometric to some UltV , where V is dually well-ordered; ii. M is point-homogeneous, the tree (Ner v(M ), ⊇) is well founded and the degree of every nonmaximal element of it is infinite; iii. M is homogeneous and indivisible. Let M be an ultrametric space and B ∈ Nerv(M) of diameter r. Then the relation d(x, y) < r on B is an equivalence relation. The equivalence classes of this relation are the sons of B. Lemma 2.1. Let λ be a countable chain and ν ∈ λ and a a degree function for λ with a(ν) = n ∈ ω and w : λ ∪ {∞} → R+ a strictly decreasing map with w(∞) = 0 and image V . Let dw := w ◦ ∆ and let M = (ω[a] , dw ) be the corresponding ultrametric space with Spec(M) = V . Then M is age divisible (that is, not age indivisible).
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Proof. The metric space A with |A| = n and d(x, y) = w(ν) for all x, y ∈ A with x ̸= y is an element in the age of M. Let R be an element in the age of M. The balls in R of diameter w(ν) are disjoint and each contains at most n ≥ 2 sons. Let P be the subset of R containing the elements of exactly one son from every ball of R which contains n elements. Then A cannot be embedded into R ↓ P and it also cannot be embedded into R ↓ (R \ P ). Lemma 2.2. Let λ be a countable chain and w : λ∪{∞} → R+ a strictly decreasing map with w(∞) = 0 and image V . Let dw := w ◦ ∆. Then the corresponding ultrametric space M = (ω[λ] , dw ) is weakly indivisible. Proof. For a function f and a set S let f ′′ (S ) = {f (s) | s ∈ S , f (s) defined}. Let r ∈ V . Observe that the balls in M of diameter r are pairwise disjoint and cover M. If B is a ball of diameter r with x, y ∈ B and if z ∈ M but z ̸∈ B then d(x, z ) = d(y, z ) > r ≥ d(x, y). This implies that if for every ball B of diameter r there is an embedding eB of B into B then the union of those embeddings eB is an embedding of M into M. Every ball B of diameter r has infinitely many sons. Hence if S is a set of points in M which contains points of only finitely many sons of every ball B of diameter r then there exists an embedding f of M into M with f (x) ̸∈ S for every point x of M. Let A be an element in the age of M and P ⊆ M such that A cannot be embedded into M ↓ P. We will show by induction on |A| that then M can be embedded into M ↓ (M \ P ). The lemma obviously holds in the case |A| = 1. Let r be the smallest distance between two different elements of A and K a ball in A of diameter r. Note that |K | > 1 and that any two different elements in K have distance r. Let p, q ∈ K with q ̸= p. Let F be the set of isometries of A \ {p} into P. Then F ̸= ∅; otherwise M could be embedded into M ↓ (M \ P ) because of the induction assumption. Let B be the set of balls B of diameter r for which there is an f ∈ F with f (q) ∈ B. It follows that f ′′ (K \ {p}) ⊆ B contains points from |K | − 1 sons of B. If P contains points of more than |K | − 1 sons of B then there is an element x ∈ P \ f ′′ (K \ {p}) such that the space A can be embedded into M ↓ (P ∪ {x}). It follows that P ∩ B contains points from only finitely many sons of B. If S = {f (q) | f ∈ F } then there is no isometry of A \ {p} into P \ S. Note that S ∩ B contains points from only finitely many sons of B for every ball B of M of diameter r. Using induction, there is an embedding g of M into M ↓ ((M \ P ) ∪ S ). Let M∗ be the image of g. Then S contains for every ball B of M∗ of diameter r points from only finitely many sons of B and hence there exists an embedding h of M∗ into M∗ with h(x) ̸∈ S for all points x of M∗ . It follows that the composition h ◦ g is an embedding of M into M ↓ (M \ P ). Theorem 2.3. Let M be a denumerable ultrametric space. Then the following properties are equivalent: i. M is isometric to some UltV . ii. M is point-homogeneous and the degree of every nonmaximal element of the tree (Ner v(M ), ⊇) is infinite. iii. M is homogeneous and weakly indivisible. Proof. Follows from Theorem 2.1 and Lemma 2.2.
Corollary 2.1. A denumerable homogeneous ultrametric space is age indivisible if and only if it is weakly indivisible. Proof. Follows from Theorem 2.3 and Lemma 2.1.
3. A sufficient condition For F a finite subset of elements of a homogeneous structure H and a ∈ H \ F we denote by orbit(F , a) the set of elements b ∈ H for which there exists an automorphism g of H with g (a) = b and g (x) = x for all x ∈ F . The orbits of H are all subsets of H of the form orbit(F , a) with F ⊆ H finite and a ∈ H \ F . Note that if every orbit of H induces a substructure isomorphic to H then H is indivisible. (Enumerate H into an ω-sequence and observe that for every partition of H into two parts (P , H \ P ) we can find step by step an embedding of H into H ↓ P or if we are obstructed from doing so, then one of the orbits is a subset of H \ P.)
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Theorem 3.1. Let H be a countable homogeneous structure such that for every finite subset F ⊆ H and a ∈ H \ F and element V in the age of H and element v ∈ V there exists a copy K of H with K ⊆ H \ F such that every embedding e of V ↓ (V \ {v}) into K has an extension e∗ to an embedding of V into H with e∗ (v) ∈ orbit(F , a). Then H is weakly indivisible. Proof. Let V be an element in the age of H and let (P , H \ P ) be a partition of H such that there is no embedding of V into H ↓ (H \ P ). We will prove by induction on |V | that there is an embedding of H into P. If |V | = 1 the assertion obviously holds. Let v ∈ V and U = V ↓ (V \ {v}). Let h0 , h1 , h2 , h3 , . . . be an ω-enumeration of H. If there is a sequence p0 , p1 , p2 , p3 , . . . of elements in P such that the map f : H → P with f (hi ) = pi for all i ∈ ω is an embedding, we are done. Otherwise there exists a finite sequence p0 , p1 , p2 , p3 , . . . , pn−1 of elements in P such that the function f mapping hi to pi for i ∈ n is a local isomorphism which cannot be extended to a local isomorphism mapping hn into P. Because H has the mapping extension property there exists an element a ∈ H such that the function f can be extended to a local isomorphism mapping hn to an element a ∈ H. Let F := {pi | i ∈ n} and note that orbit(F , a) ∩ P = ∅ because f cannot be extended to a function mapping hn to an element in P. Because K is an isomorphic copy of H and (K ∩ P , K \ P ) is a partition of K it follows from the induction assumption that there is an embedding e of V ↓ (V \ {v}) into K ↓ (K \ P ) which according to the assumption of the theorem has an extension e∗ to an embedding of V into H with e∗ (v) ∈ orbit(F , a). We have arrived at a contradiction. As observed earlier, orbit(F , a) ∩ P = ∅ and hence e∗ (v) ̸∈ P and hence in H \ P and e∗ is an embedding of V into H ↓ (H \ P ). 3.1. Free amalgamation Let A be a relational structure. The elements x and y in A are adjacent if x ̸= y and there exists an n ∈ ω and an n-ary relation R in the signature of A and an n-tuple (z0 , z1 , . . . , zn−1 ) with entries which contain x and y and are elements of A and with R(z0 , z1 , . . . , zn ). The relational structure A is complete if any two different elements of it are adjacent. Let A and B be two structures with the same signature. The free amalgam of A and B is the structure D with D = A ∪ B and D ↓ A = A and D ↓ B = B and if x ∈ A \ B and y ∈ B \ A then x and y are not adjacent in D. Note that in the free amalgam D of A and B no point in A \ B is identified with a point in B \ A. An age has free amalgamation if any two of its elements have free amalgamation. Note that if a countable age has free amalgamation then it has amalgamation and hence there exists a unique countable homogeneous structure with that age. An age has free amalgamation over the empty set if any two disjoint elements of the age have free amalgamation. An age has free vertex amalgamation if any two of its elements whose intersection is a singleton have free amalgamation. The age A has free amalgamation if and only if there exists a set B of finite complete structures in the signature σ of the elements of A such that A is the class of all finite structures in signature σ which do not embed any of the elements in B; see [17]. Lemma 3.1. Let A be an infinite countable age which has free vertex amalgamation. Then the age of H has free empty set amalgamation. Proof. Let A and B be two elements in the age of H with A ∩ B = ∅. According to Fraïssé’s theorem (see [6]) there exists a countable infinite structure H whose age is equal to A. We may assume that H ∩ (A ∪ B) = ∅. There exist copies A∗ of A and B∗ of B in H. Because H is infinite there is a point p ∈ H with p ̸∈ A∗ ∪ B∗ . Let A′ = H ↓ ({p} ∪ A∗ ) and B′ = H ↓ ({p} ∪ B∗ ). Let B′′ be the structure with B′′ = {p} ∪ B for which there exists an isomorphism f : B′′ → B′ with f (p) = p. Let A′′ be the structure with A′′ = {p} ∪ A for which there exists an isomorphism f : A′′ → A′ with f (p) = p. It follows that A′′ ∩ B′′ = {p} and hence there exists a free amalgam D′′ of A′′ and B′′ . The restriction of D′′ to A ∪ B is a free amalgam of A and B.
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Note: Let H be the structure which consists of a single point and no relations. The age of H has, leave vacuously, free vertex amalgamation but does not have free amalgamation over the empty set. Let L be the countable structure with countably many unary relations. Each point in L is in only one of the relations and different points are in different unary relations. The age of L does not have free vertex amalgamation. Indeed, it is not weakly indivisible—actually not even age indivisible. More generally, the age of a structure which has only finitely points of a given singleton type does not have free vertex amalgamation. Lemma 3.2. Let H be a countable homogeneous structure whose age has free amalgamation over the empty set. Then there exists for every finite subset F of H a copy K of H in H with K ⊆ H \ F and such that H ↓ (F ∪ K ) is the free amalgam of H ↓ F and K. Proof. Let H ↓ F := F and C be an isomorphic copy of F with C ∩ H = ∅. Let D be a free amalgam of F with C. Then D is an element of the age of H and hence there is an embedding f of D into H with f (x) = x for all x ∈ F . Let K := f ′′ (C ). Theorem 3.2. Let H be an infinite countable homogeneous structure with free vertex amalgamation. Then H is weakly indivisible. Proof. Using Theorem 3.1. Let F be a finite subset of H and a ∈ H \ F and V be in the age of H and v ∈ V . Let K be a copy of H in H with K ⊆ H \ F and such that H ↓ (F ∪ K ) is the free amalgam of H ↓ F and K; this exists according to Lemmas 3.1 and 3.2. Let a ∈ H \ F and G := H ↓ (F ∪ {a}) and W be a copy of V ↓ (V \ {v}) in K with e an embedding V ↓ (V \ {v}) with image W. Let D be the free amalgam of G with V identifying a with v and let e′ be the embedding of D ↓ (F ∪ V \ {v}) into H with e′ (x) = x for all x ∈ F and e′ (x) = e(x) for all x ∈ V \ {v}. Then e′ has an extension to an embedding e∗ of D into H with e∗ (v) ∈ orbit(F , a). By graph, we mean simple graph, that is with no loops or multiple edges. Corollary 3.1. Every countable infinite homogeneous graph G is weakly indivisible. Proof. Note that a graph is weakly indivisible if and only if its complement is weakly indivisible and that a graph is age indivisible if and only if the age of its complement is weakly indivisible. The Kn -free homogeneous graphs have free amalgamation and hence are weakly indivisible; they are actually indivisible according to [4]. The Rado graph has free amalgamation and it is easy to see that it is indivisible. (Every orbit of the Rado graph induces a subgraph isomorphic to the Rado graph.) Otherwise G is a union of complete graphs and hence weakly indivisible. The corollary follows from the Lachlan and Woodrow classification of countable homogeneous graphs; see [10]. An example of a weakly indivisible but not indivisible homogeneous structure: Let H be a countable homogeneous structure with binary signature. The orbits of H form a partial order under embedding. If H is indivisible then this partial order is a chain; see [18]. Let A be the class of all finite structures of the form (V ; E0 , E1 ) where (V ; E0 ) is a triangle free graph with vertex set V and edge set E0 and (V ; E1 ) is a triangle free graph with vertex set V and edge set E1 and E0 ∩ E1 = ∅. Then A is a countable age with free amalgamation. It follows from Theorem 3.2 that the countable homogeneous graph HA whose age is A is weakly indivisible. Let x ∈ H. The orbit of HA induced by all vertices adjacent to x by an edge in E0 is the Rado graph in edge set E1 . The orbit of HA induced by all vertices adjacent to x by an edge in E1 is the Rado graph in edge set E0 . Hence the partial order of orbits under embedding is not a chain. It follows that HA is not indivisible. 3.2. Oriented graphs We will use Lachlan’s classification of the countable homogeneous tournaments (see [9]) and the classification of Cherlin of the countable homogeneous directed graphs (see [1]).
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Let L be a dense countable subset of the unit circle such that no two of its elements are opposite on the circle and such that L contains the point (1, 0). The oriented graph L with vertex set L has an arc from x to y if x ̸= y and the counterclockwise angle from x to y is less than π . The graph L is the local order. The random tournament, the order structure of the rationals and the local order are the three countable infinite homogeneous tournaments. The random tournament and the order structure of the rationals are easily seen to be divisible and hence weakly indivisible. Let L0 be the set of elements in L whose angle with (1, 0) is smaller than π and L1 = L \ L0 . The partition of L into L0 and L1 shows that L is not age indivisible. Note that the tournaments are the complete structures amongst the oriented graphs. Let T be a set of finite tournaments and AT be the set of finite oriented graphs which do not embed any of the tournaments in T and HT be the countable homogeneous oriented graph whose age is AT . Then HT has free amalgamation. Hence all the ‘‘Henson graphs’’ (see [8]) are weakly indivisible according to Theorem 3.1. The countable infinite oriented graph with no edges is weakly indivisible. The random oriented graph has free amalgamation and hence is weakly indivisible. Let C be a dense countable subset of the unit circle such that no two of its elements have a counterclockwise angle of 2π /3 and such that C contains the point (1, 0). The oriented graph C with vertex set C has an arc from x to y if x ̸= y and the counterclockwise angle from x to y is less than 2π /3. The graph C is the circular order. Let C0 be the set of elements in C whose angle with (1, 0) is smaller than 2π /3 and C1 be the set of elements in C whose angle with (1, 0) is in between 2π /3 and 4π /3, and C2 := C \ (C0 ∪ C1 ). The partition of C into C0 and C1 and C2 shows that L is not age indivisible. Let n ∈ ω and An be the age of finite oriented graphs with no n-element independent set and Hn be the corresponding countably infinite homogeneous oriented graph whose age is An . Then Hn is weakly indivisible on account of Theorem 3.1 as follows: Let F be a finite subset of Hn and a ∈ Hn \ F and V ∈ An and v ∈ V . Let K be the set of all elements in Hn \ F such that there is an arc from x to y for every element x ∈ F and y ∈ K . Then Hn ↓ K is a copy of Hn in Hn . It is easily seen that K satisfies the conditions of Theorem 3.1. The random partial order is indivisible because each of its orbits is an isomorphic copy of it; hence it is weakly indivisible. There are several more such countable homogeneous oriented graphs and of course many other special cases of countable homogeneous structures. I could not detect a likely candidate which is age indivisible but not weakly indivisible. The best open case is the rational Urysohn sphere as mentioned in the introduction. In general, metric spaces may provide such examples as all of the Urysohn spaces are age indivisible, which follows by an easy application of the Hales–Jewett theorem; see [7]. Acknowledgment The author was supported by NSERC of Canada Grant # 691325. References [1] G. Cherlin, The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments, Memoirs of the American Mathematical Society 131 (621) (1998) xiv+161. [2] C. Delhommé, C. Laflamme, M. Pouzet, N. Sauer, Divisibility of countable metric spaces, European Journal of Combinatorics 28 (6) (2007) 1746–1769. [3] C. Delhommé, C. Laflamme, M. Pouzet, N. Sauer, Indivisible ultrametric spaces, in: Workshop on the Urysohn Space, Topology and its Applications 155 (14) (2008) 1462–1478 (special issue). [4] M. El-Zahar, N. Sauer, The Indivisibility of the homogeneous Kn -free graphs, Journal of Combinatorial Theory, Series B 47 (2) (1989) 162–170. [5] M. El-Zahar, N. Sauer, Ramsey-type properties of relational structures, Discrete Mathematics 94 (1991) 1–10. [6] R. Fraïssé, Theory of Relations, in: Studies in Logic and the Foundations of Mathematics, vol. 145, North-Holland Publishing Co., Amsterdam, 2000. [7] A.W. Hales, R.I. Jewett, Regularity and positional games, Transactions of the American Mathematical Society 106 (1963) 222–229. [8] C.W. Henson, Countable homogeneous relational structures and ℵ0 -categorical theories, Journal of Symbolic Logic 37 (1972) 494–500. [9] A.H. Lachlan, Countable homogeneous tournaments, Transactions of the American Mathematical Society 284 (2) (1984) 431–461.
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[10] A.H. Lachlan, R.E. Woodrow, Countable ultrahomogeneous undirected graphs, Transactions of the American Mathematical Society 262 (1980) 51–94. [11] C. Laflamme, L. Nguyen Van Thé, N. Sauer, Partition properties of the dense local order and a colored version of Milliken’s theorem, Combinatorica 30 (1) (2010) 83–104. [12] L. Nguyen Van Thé, N. Sauer, The Urysohn sphere is oscillation stable, Goemetric and Functional Analysis 19 (2) (2009) 536–557. [13] L. Nguyen Van Thé, N. Sauer, Some weak indivisibility results in ultrahomogeneous metric spaces, European Journal of Combinatorics 31 (5) (2010) 1464–1483. [14] M. Pouzet, Relation minimale, Zeitschrift für Mathematische Logik 25 (1979) 315–344. fasc. 4. [15] M. Pouzet, Relation impartible, Dissertationnes 103 (1981) 1–48. [16] M. Pouzet, C. Laflamme, L. Nguyen Van Thé, Partitions and indivisibility properties of countable dimensional vector spaces, Journal of Combinatorial Theory Series A 118 (1) (2011) 67–77. [17] N. Sauer, in: R. Fraïssé (Ed.), Appendix in: Theory of Relations, revised ed., in: Studies in Logic and the Foundations of Mathematics, vol. 145, North-Holland Publishing Co., Amsterdam, 2000. [18] N. Sauer, Canonical vertex partitions, Combinatorics Probability and Computing 12 (5–6) (2003) 671–704. [19] N. Sauer, Vertex partitions of metric spaces with finite distance sets, Discrete Mathematics 312 (1) (2012) 119–128.
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Age and weak indivisibility N. Sauer University of Calgary, Department of Mathematics and Statistics, University Drive 2400, Calgary, T2N 1N4, Alberta, Canada Institute of Discrete Mathematics and Geometry, University of Technology in Wien (Vienna), Austria
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Article history: Available online 9 August 2013
abstract A relational structure H is homogeneous if every isomorphism between finite induced substructures has an extension to an automorphism of H. A relational structure R is indivisible if for every partition (A, B) of the set of elements of R, there is a copy of R in A or in B. The structure R is weakly indivisible if for every partition (A, B) of the set of elements of R for which some finite induced substructure of R does not have a copy in A, there exists a copy of R in B. The structure R is age indivisible if for every partition (A, B) of the elements of R every finite induced substructure of R has an embedding into A or every finite induced substructure of R has an embedding into B. It follows that indivisibility implies weak indivisibility which in turn implies age indivisibility. There are many examples of countable infinite homogeneous structures which are weakly indivisible but not indivisible; see Sauer (2000) [17] and the end of Section 3.1 of this paper. In general, it is difficult to find structures which are age indivisible but not weakly indivisible. One such example (see Pouzet et al. (2011) [16]) is obtained via the inhomogeneous general linear group of vector spaces over finite fields. Finding such an example for homogeneous structures seems to be even more difficult. The only one that I obtained, together with L. Nguyen Van Thé, is a countable homogeneous metric subspace of the Hilbert sphere ℓ2 (see Nguyen Van Thé and Sauer (2010) [13]). Both of the examples above are relational structures with infinitely many relations. One of the still open questions then is: Are there countable infinite homogeneous structures with finite signature which are age but not weakly indivisible? We will show that the generic ‘‘free amalgamation’’ homogeneous structures are weakly indivisible; see Section 3.1. Countable ‘‘Urysohn metric spaces’’ (see the next section), with finite distance sets, are indivisible (see Sauer (2013) [19]).
E-mail address: [email protected]. 0195-6698/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ejc.2013.07.013
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Countable Urysohn metric spaces are age indivisible, which follows from the Hales–Jewett theorem (Hales and Jewett (1963) [7]; see Delhommé et al. (2007) [2]). For non-Urysohn homogeneous countable metric spaces, except for the example mentioned above, the weak indivisibility question is completely open. We will show, in this paper, that a countable homogeneous ultrametric space is age indivisible if and only if it is weakly indivisible. © 2013 Elsevier Ltd. All rights reserved.
1. Basic notions See [6,17] for a more complete introduction to homogeneous structures and the theory of partitions of such structures. Relational structures are denoted with capital Roman letters and the same letter in slanted format (italics) is used to indicate the corresponding base set. Let R be a relational structure and S ⊆ R. Then R ↓ S denotes the restriction of R to S, that is the substructure of R induced by S. A copy of the relational structure R in the relational structure B is an induced substructure of B isomorphic to R. The relational structure R can be embedded into the structure B if it has a copy in B. The age of R consists of all finite structures which can be embedded into R. A countable set of finite relational structures having the same signature is an age if it is closed under isomorphism and induced substructures and is updirected, that is for any two structures in the age there is one into which both can be embedded. An age is infinite if it has infinitely many isomorphism types and countable if it has countably many isomorphism types. For every countable age A there is a countable structure R whose age is A; see [6]. An age A is a vertex Ramsey class if for every structure R ∈ A there is a structure S ∈ A such that for every partition (A, B) of S there is a copy of R in A or in B. A local isomorphism of A is an isomorphism of a finite induced substructure of A to an induced substructure of A. A relational structure H is homogeneous if every local isomorphism of H has an extension to an automorphism of H. Let R be a countable relational structure; then R is: 1. Age indivisible if for every partition (A, B) of the elements of R every finite induced substructure of R has an embedding into A or every finite induced substructure of R has an embedding into B. 2. Weakly indivisible if for every partition (A, B) of R for which there exists an element A in the age of R which cannot be embedded into R ↓ A, the structure R can be embedded into R ↓ B. 3. Indivisible if for every partition (A, B) of R there is a copy of R in A or in B. A countable relational structure R is age indivisible if and only if its age is a vertex Ramsey class; for a proof see [5]. For some earlier results on indivisible ages and indivisible relations by M. Pouzet, see [14,15]. There are two large classes of prominent countable infinite homogeneous structures. One of them is the class of homogeneous structures whose age has free amalgamation; see Section 3.1. According to Theorem 3.2 all of those homogeneous structures are weakly indivisible. For most signatures a complete classification of homogeneous structures whose age does not have free amalgamation is completely beyond our abilities. There are some sporadic cases such as the countable infinite graph whose every connected component is a triangle, or more generally the complete graph on n vertices for a fixed n. Of course in such cases the divisibility problems can easily be resolved. On the other hand to resolve the partition question for the dense local order has not been easy; see [11]. For the case of vector spaces some special cases are discussed in [16]. The other one is the class of countable homogeneous metric spaces. An Urysohn metric space with V as set of distances is a homogeneous complete metric space which embeds every separable, and hence every finite, metric space F whose set of distances is a subset of V , the classical Urysohn metric space being the unique complete separable metric space which embeds every separable metric space—in this case V being the set of nonnegative reals. Countable Urysohn metric spaces are age indivisible;
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see [2]. Unfortunately we cannot decide whether such spaces are weakly indivisible, not even for the most prominent one, the rational Urysohn sphere, that is for the countable Urysohn space whose set of distances is the interval [0, 1] ∩ Q. We can only prove that it is ‘‘approximately weakly indivisible’’; see [13]. Countable Urysohn metric spaces with a finite set of distances are indivisible (see [19]) and the countable Urysohn metric space with distance set the nonnegative integers is weakly indivisible (see [12]). Corollary 2.1 settles the case of countable ultrametric spaces: a denumerable homogeneous ultrametric space is weakly indivisible if and only if it is age indivisible. 2. Ultrahomogeneous metric spaces Let f be a function of a set S to ω. Then supp(f ) := {s ∈ S : f (s) ̸= 0}. Let λ be a countable chain, that is a linearly ordered countable set. A function a : λ → ω with 2 ≤ a(µ) ≤ ω for all µ ∈ λ will be called a degree function for λ. Let a be a degree function for λ; then
ω[a] := {f : λ → ω | ∀µ ∈ λ (f (µ) < a(µ)) and supp(f ) is finite}. So for example let λ = ω and a(n) = 2 for all n ∈ ω. Then every f ∈ ω[a] is a function from ω into {0, 1} := 2 for which there are only finitely many n ∈ ω with f (n) = 1 and with f (n) = 0 or f (n) = 1 for all n ∈ ω. It follows that ω[a] is a subtree of the Cantor tree, that is the tree of height ω whose branches are the set of all 0, 1-sequences. In this case, for the distance ∆ defined below, we have that ∆(f , g ) is the smallest level of the tree, that is the smallest n ∈ ω, for which f (n) ̸= g (n). Associate with every level of the tree, that is with every positive integer n, the number 1n . Let d(f , g ) := 1n with n = ∆(f , g ). Then it turns out (see Theorem 2.1 below) that ω[a] in this case is an ultrametric homogeneous space with distance function d. If a(µ) = ω for every µ ∈ λ, then the set ω[a] is usually denoted by ω[λ] . For f , g ∈ ω[a] let
∆(f , g ) :=
∞, µ,
if f = g ; for µ the smallest element in λ with f (µ) ̸= g (µ).
For M a metric space and a ∈ M let Spec(M, a) := {d(a, x) | x ∈ M } and Spec(M) := {d(x, y) | x, y ∈ M } and Ba (s) := {x ∈ M | d(a, x) ≤ s} and Nerv(M ) := {Ba (s) : a ∈ M , s ∈ Spec(M , a)}. An ultrametric space M is a metric space with d(x, y) ≤ max{d(x, z ), d(y, z )} for all x, y, z ∈ M. The following theorems, Theorems 2.1 and 2.2, are from [2]; see [3] as well. Theorem 2.1. Let λ be a countable chain and a a degree function for λ and w : λ ∪ {∞} → R+ a strictly decreasing map with w(∞) = 0 and image V . Let dw := w ◦ ∆. Then M = (ω[a] , dw ) is an ultrametric space with Spec(M) = V . Let M be a countable ultrametric space. Then the following properties are equivalent: a. M is isometric to some (ω[a] , dw ). b. M is homogeneous. c. M is point-homogeneous. (That is, the automorphism group Iso(M) of M acts transitively on M.) The space (ω[λ] , dw ) is the countable homogeneous ultrametric space UltV associated with V . Theorem 2.2. Let M be a denumerable ultrametric space. Then the following properties are equivalent: i. M is isometric to some UltV , where V is dually well-ordered; ii. M is point-homogeneous, the tree (Ner v(M ), ⊇) is well founded and the degree of every nonmaximal element of it is infinite; iii. M is homogeneous and indivisible. Let M be an ultrametric space and B ∈ Nerv(M) of diameter r. Then the relation d(x, y) < r on B is an equivalence relation. The equivalence classes of this relation are the sons of B. Lemma 2.1. Let λ be a countable chain and ν ∈ λ and a a degree function for λ with a(ν) = n ∈ ω and w : λ ∪ {∞} → R+ a strictly decreasing map with w(∞) = 0 and image V . Let dw := w ◦ ∆ and let M = (ω[a] , dw ) be the corresponding ultrametric space with Spec(M) = V . Then M is age divisible (that is, not age indivisible).
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Proof. The metric space A with |A| = n and d(x, y) = w(ν) for all x, y ∈ A with x ̸= y is an element in the age of M. Let R be an element in the age of M. The balls in R of diameter w(ν) are disjoint and each contains at most n ≥ 2 sons. Let P be the subset of R containing the elements of exactly one son from every ball of R which contains n elements. Then A cannot be embedded into R ↓ P and it also cannot be embedded into R ↓ (R \ P ). Lemma 2.2. Let λ be a countable chain and w : λ∪{∞} → R+ a strictly decreasing map with w(∞) = 0 and image V . Let dw := w ◦ ∆. Then the corresponding ultrametric space M = (ω[λ] , dw ) is weakly indivisible. Proof. For a function f and a set S let f ′′ (S ) = {f (s) | s ∈ S , f (s) defined}. Let r ∈ V . Observe that the balls in M of diameter r are pairwise disjoint and cover M. If B is a ball of diameter r with x, y ∈ B and if z ∈ M but z ̸∈ B then d(x, z ) = d(y, z ) > r ≥ d(x, y). This implies that if for every ball B of diameter r there is an embedding eB of B into B then the union of those embeddings eB is an embedding of M into M. Every ball B of diameter r has infinitely many sons. Hence if S is a set of points in M which contains points of only finitely many sons of every ball B of diameter r then there exists an embedding f of M into M with f (x) ̸∈ S for every point x of M. Let A be an element in the age of M and P ⊆ M such that A cannot be embedded into M ↓ P. We will show by induction on |A| that then M can be embedded into M ↓ (M \ P ). The lemma obviously holds in the case |A| = 1. Let r be the smallest distance between two different elements of A and K a ball in A of diameter r. Note that |K | > 1 and that any two different elements in K have distance r. Let p, q ∈ K with q ̸= p. Let F be the set of isometries of A \ {p} into P. Then F ̸= ∅; otherwise M could be embedded into M ↓ (M \ P ) because of the induction assumption. Let B be the set of balls B of diameter r for which there is an f ∈ F with f (q) ∈ B. It follows that f ′′ (K \ {p}) ⊆ B contains points from |K | − 1 sons of B. If P contains points of more than |K | − 1 sons of B then there is an element x ∈ P \ f ′′ (K \ {p}) such that the space A can be embedded into M ↓ (P ∪ {x}). It follows that P ∩ B contains points from only finitely many sons of B. If S = {f (q) | f ∈ F } then there is no isometry of A \ {p} into P \ S. Note that S ∩ B contains points from only finitely many sons of B for every ball B of M of diameter r. Using induction, there is an embedding g of M into M ↓ ((M \ P ) ∪ S ). Let M∗ be the image of g. Then S contains for every ball B of M∗ of diameter r points from only finitely many sons of B and hence there exists an embedding h of M∗ into M∗ with h(x) ̸∈ S for all points x of M∗ . It follows that the composition h ◦ g is an embedding of M into M ↓ (M \ P ). Theorem 2.3. Let M be a denumerable ultrametric space. Then the following properties are equivalent: i. M is isometric to some UltV . ii. M is point-homogeneous and the degree of every nonmaximal element of the tree (Ner v(M ), ⊇) is infinite. iii. M is homogeneous and weakly indivisible. Proof. Follows from Theorem 2.1 and Lemma 2.2.
Corollary 2.1. A denumerable homogeneous ultrametric space is age indivisible if and only if it is weakly indivisible. Proof. Follows from Theorem 2.3 and Lemma 2.1.
3. A sufficient condition For F a finite subset of elements of a homogeneous structure H and a ∈ H \ F we denote by orbit(F , a) the set of elements b ∈ H for which there exists an automorphism g of H with g (a) = b and g (x) = x for all x ∈ F . The orbits of H are all subsets of H of the form orbit(F , a) with F ⊆ H finite and a ∈ H \ F . Note that if every orbit of H induces a substructure isomorphic to H then H is indivisible. (Enumerate H into an ω-sequence and observe that for every partition of H into two parts (P , H \ P ) we can find step by step an embedding of H into H ↓ P or if we are obstructed from doing so, then one of the orbits is a subset of H \ P.)
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Theorem 3.1. Let H be a countable homogeneous structure such that for every finite subset F ⊆ H and a ∈ H \ F and element V in the age of H and element v ∈ V there exists a copy K of H with K ⊆ H \ F such that every embedding e of V ↓ (V \ {v}) into K has an extension e∗ to an embedding of V into H with e∗ (v) ∈ orbit(F , a). Then H is weakly indivisible. Proof. Let V be an element in the age of H and let (P , H \ P ) be a partition of H such that there is no embedding of V into H ↓ (H \ P ). We will prove by induction on |V | that there is an embedding of H into P. If |V | = 1 the assertion obviously holds. Let v ∈ V and U = V ↓ (V \ {v}). Let h0 , h1 , h2 , h3 , . . . be an ω-enumeration of H. If there is a sequence p0 , p1 , p2 , p3 , . . . of elements in P such that the map f : H → P with f (hi ) = pi for all i ∈ ω is an embedding, we are done. Otherwise there exists a finite sequence p0 , p1 , p2 , p3 , . . . , pn−1 of elements in P such that the function f mapping hi to pi for i ∈ n is a local isomorphism which cannot be extended to a local isomorphism mapping hn into P. Because H has the mapping extension property there exists an element a ∈ H such that the function f can be extended to a local isomorphism mapping hn to an element a ∈ H. Let F := {pi | i ∈ n} and note that orbit(F , a) ∩ P = ∅ because f cannot be extended to a function mapping hn to an element in P. Because K is an isomorphic copy of H and (K ∩ P , K \ P ) is a partition of K it follows from the induction assumption that there is an embedding e of V ↓ (V \ {v}) into K ↓ (K \ P ) which according to the assumption of the theorem has an extension e∗ to an embedding of V into H with e∗ (v) ∈ orbit(F , a). We have arrived at a contradiction. As observed earlier, orbit(F , a) ∩ P = ∅ and hence e∗ (v) ̸∈ P and hence in H \ P and e∗ is an embedding of V into H ↓ (H \ P ). 3.1. Free amalgamation Let A be a relational structure. The elements x and y in A are adjacent if x ̸= y and there exists an n ∈ ω and an n-ary relation R in the signature of A and an n-tuple (z0 , z1 , . . . , zn−1 ) with entries which contain x and y and are elements of A and with R(z0 , z1 , . . . , zn ). The relational structure A is complete if any two different elements of it are adjacent. Let A and B be two structures with the same signature. The free amalgam of A and B is the structure D with D = A ∪ B and D ↓ A = A and D ↓ B = B and if x ∈ A \ B and y ∈ B \ A then x and y are not adjacent in D. Note that in the free amalgam D of A and B no point in A \ B is identified with a point in B \ A. An age has free amalgamation if any two of its elements have free amalgamation. Note that if a countable age has free amalgamation then it has amalgamation and hence there exists a unique countable homogeneous structure with that age. An age has free amalgamation over the empty set if any two disjoint elements of the age have free amalgamation. An age has free vertex amalgamation if any two of its elements whose intersection is a singleton have free amalgamation. The age A has free amalgamation if and only if there exists a set B of finite complete structures in the signature σ of the elements of A such that A is the class of all finite structures in signature σ which do not embed any of the elements in B; see [17]. Lemma 3.1. Let A be an infinite countable age which has free vertex amalgamation. Then the age of H has free empty set amalgamation. Proof. Let A and B be two elements in the age of H with A ∩ B = ∅. According to Fraïssé’s theorem (see [6]) there exists a countable infinite structure H whose age is equal to A. We may assume that H ∩ (A ∪ B) = ∅. There exist copies A∗ of A and B∗ of B in H. Because H is infinite there is a point p ∈ H with p ̸∈ A∗ ∪ B∗ . Let A′ = H ↓ ({p} ∪ A∗ ) and B′ = H ↓ ({p} ∪ B∗ ). Let B′′ be the structure with B′′ = {p} ∪ B for which there exists an isomorphism f : B′′ → B′ with f (p) = p. Let A′′ be the structure with A′′ = {p} ∪ A for which there exists an isomorphism f : A′′ → A′ with f (p) = p. It follows that A′′ ∩ B′′ = {p} and hence there exists a free amalgam D′′ of A′′ and B′′ . The restriction of D′′ to A ∪ B is a free amalgam of A and B.
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Note: Let H be the structure which consists of a single point and no relations. The age of H has, leave vacuously, free vertex amalgamation but does not have free amalgamation over the empty set. Let L be the countable structure with countably many unary relations. Each point in L is in only one of the relations and different points are in different unary relations. The age of L does not have free vertex amalgamation. Indeed, it is not weakly indivisible—actually not even age indivisible. More generally, the age of a structure which has only finitely points of a given singleton type does not have free vertex amalgamation. Lemma 3.2. Let H be a countable homogeneous structure whose age has free amalgamation over the empty set. Then there exists for every finite subset F of H a copy K of H in H with K ⊆ H \ F and such that H ↓ (F ∪ K ) is the free amalgam of H ↓ F and K. Proof. Let H ↓ F := F and C be an isomorphic copy of F with C ∩ H = ∅. Let D be a free amalgam of F with C. Then D is an element of the age of H and hence there is an embedding f of D into H with f (x) = x for all x ∈ F . Let K := f ′′ (C ). Theorem 3.2. Let H be an infinite countable homogeneous structure with free vertex amalgamation. Then H is weakly indivisible. Proof. Using Theorem 3.1. Let F be a finite subset of H and a ∈ H \ F and V be in the age of H and v ∈ V . Let K be a copy of H in H with K ⊆ H \ F and such that H ↓ (F ∪ K ) is the free amalgam of H ↓ F and K; this exists according to Lemmas 3.1 and 3.2. Let a ∈ H \ F and G := H ↓ (F ∪ {a}) and W be a copy of V ↓ (V \ {v}) in K with e an embedding V ↓ (V \ {v}) with image W. Let D be the free amalgam of G with V identifying a with v and let e′ be the embedding of D ↓ (F ∪ V \ {v}) into H with e′ (x) = x for all x ∈ F and e′ (x) = e(x) for all x ∈ V \ {v}. Then e′ has an extension to an embedding e∗ of D into H with e∗ (v) ∈ orbit(F , a). By graph, we mean simple graph, that is with no loops or multiple edges. Corollary 3.1. Every countable infinite homogeneous graph G is weakly indivisible. Proof. Note that a graph is weakly indivisible if and only if its complement is weakly indivisible and that a graph is age indivisible if and only if the age of its complement is weakly indivisible. The Kn -free homogeneous graphs have free amalgamation and hence are weakly indivisible; they are actually indivisible according to [4]. The Rado graph has free amalgamation and it is easy to see that it is indivisible. (Every orbit of the Rado graph induces a subgraph isomorphic to the Rado graph.) Otherwise G is a union of complete graphs and hence weakly indivisible. The corollary follows from the Lachlan and Woodrow classification of countable homogeneous graphs; see [10]. An example of a weakly indivisible but not indivisible homogeneous structure: Let H be a countable homogeneous structure with binary signature. The orbits of H form a partial order under embedding. If H is indivisible then this partial order is a chain; see [18]. Let A be the class of all finite structures of the form (V ; E0 , E1 ) where (V ; E0 ) is a triangle free graph with vertex set V and edge set E0 and (V ; E1 ) is a triangle free graph with vertex set V and edge set E1 and E0 ∩ E1 = ∅. Then A is a countable age with free amalgamation. It follows from Theorem 3.2 that the countable homogeneous graph HA whose age is A is weakly indivisible. Let x ∈ H. The orbit of HA induced by all vertices adjacent to x by an edge in E0 is the Rado graph in edge set E1 . The orbit of HA induced by all vertices adjacent to x by an edge in E1 is the Rado graph in edge set E0 . Hence the partial order of orbits under embedding is not a chain. It follows that HA is not indivisible. 3.2. Oriented graphs We will use Lachlan’s classification of the countable homogeneous tournaments (see [9]) and the classification of Cherlin of the countable homogeneous directed graphs (see [1]).
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Let L be a dense countable subset of the unit circle such that no two of its elements are opposite on the circle and such that L contains the point (1, 0). The oriented graph L with vertex set L has an arc from x to y if x ̸= y and the counterclockwise angle from x to y is less than π . The graph L is the local order. The random tournament, the order structure of the rationals and the local order are the three countable infinite homogeneous tournaments. The random tournament and the order structure of the rationals are easily seen to be divisible and hence weakly indivisible. Let L0 be the set of elements in L whose angle with (1, 0) is smaller than π and L1 = L \ L0 . The partition of L into L0 and L1 shows that L is not age indivisible. Note that the tournaments are the complete structures amongst the oriented graphs. Let T be a set of finite tournaments and AT be the set of finite oriented graphs which do not embed any of the tournaments in T and HT be the countable homogeneous oriented graph whose age is AT . Then HT has free amalgamation. Hence all the ‘‘Henson graphs’’ (see [8]) are weakly indivisible according to Theorem 3.1. The countable infinite oriented graph with no edges is weakly indivisible. The random oriented graph has free amalgamation and hence is weakly indivisible. Let C be a dense countable subset of the unit circle such that no two of its elements have a counterclockwise angle of 2π /3 and such that C contains the point (1, 0). The oriented graph C with vertex set C has an arc from x to y if x ̸= y and the counterclockwise angle from x to y is less than 2π /3. The graph C is the circular order. Let C0 be the set of elements in C whose angle with (1, 0) is smaller than 2π /3 and C1 be the set of elements in C whose angle with (1, 0) is in between 2π /3 and 4π /3, and C2 := C \ (C0 ∪ C1 ). The partition of C into C0 and C1 and C2 shows that L is not age indivisible. Let n ∈ ω and An be the age of finite oriented graphs with no n-element independent set and Hn be the corresponding countably infinite homogeneous oriented graph whose age is An . Then Hn is weakly indivisible on account of Theorem 3.1 as follows: Let F be a finite subset of Hn and a ∈ Hn \ F and V ∈ An and v ∈ V . Let K be the set of all elements in Hn \ F such that there is an arc from x to y for every element x ∈ F and y ∈ K . Then Hn ↓ K is a copy of Hn in Hn . It is easily seen that K satisfies the conditions of Theorem 3.1. The random partial order is indivisible because each of its orbits is an isomorphic copy of it; hence it is weakly indivisible. There are several more such countable homogeneous oriented graphs and of course many other special cases of countable homogeneous structures. I could not detect a likely candidate which is age indivisible but not weakly indivisible. The best open case is the rational Urysohn sphere as mentioned in the introduction. In general, metric spaces may provide such examples as all of the Urysohn spaces are age indivisible, which follows by an easy application of the Hales–Jewett theorem; see [7]. Acknowledgment The author was supported by NSERC of Canada Grant # 691325. References [1] G. Cherlin, The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments, Memoirs of the American Mathematical Society 131 (621) (1998) xiv+161. [2] C. Delhommé, C. Laflamme, M. Pouzet, N. Sauer, Divisibility of countable metric spaces, European Journal of Combinatorics 28 (6) (2007) 1746–1769. [3] C. Delhommé, C. Laflamme, M. Pouzet, N. Sauer, Indivisible ultrametric spaces, in: Workshop on the Urysohn Space, Topology and its Applications 155 (14) (2008) 1462–1478 (special issue). [4] M. El-Zahar, N. Sauer, The Indivisibility of the homogeneous Kn -free graphs, Journal of Combinatorial Theory, Series B 47 (2) (1989) 162–170. [5] M. El-Zahar, N. Sauer, Ramsey-type properties of relational structures, Discrete Mathematics 94 (1991) 1–10. [6] R. Fraïssé, Theory of Relations, in: Studies in Logic and the Foundations of Mathematics, vol. 145, North-Holland Publishing Co., Amsterdam, 2000. [7] A.W. Hales, R.I. Jewett, Regularity and positional games, Transactions of the American Mathematical Society 106 (1963) 222–229. [8] C.W. Henson, Countable homogeneous relational structures and ℵ0 -categorical theories, Journal of Symbolic Logic 37 (1972) 494–500. [9] A.H. Lachlan, Countable homogeneous tournaments, Transactions of the American Mathematical Society 284 (2) (1984) 431–461.
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