Automorphism groups and Ramsey properties of sparse graphs

01/03/2018 ∙ by David M. Evans, et al. ∙ 0

We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todorčević correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory. Resolving one of the open questions in the area, we show that Hrushovski's example of an ω-categorical sparse graph has no ω-categorical expansion with extremely amenable automorphism group.

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1. Introduction

1.1. Overview

If is a natural number, we say that a graph is -sparse if, for every finite subgraph, the number of edges is bounded above by times the number of vertices. Classes of such graphs arise in model theory in Hrushovski’s predimension constructions and are an important source of counterexamples to many questions and conjectures in model-theoretic stability theory. The main aim of this paper is to study these classes from the twin viewpoints of structural Ramsey theory and topological dynamics. As we shall see, Ramsey expansions of these classes exhibit rather different behaviour from that of classes studied previously and, correspondingly, the automorphism groups of the Fraïssé limits of these classes exhibit new phenomena in topological dynamics.

The symmetric group (on a countably infinite set ) can be considered as a Polish topological group by giving it the topology of pointwise convergence. With this topology, a subgroup of is closed if and only if it is the automorphism group of a first-order structure with domain . A subgroup of is oligomorphic if has finitely many orbits on for all natural numbers . It is well known that, by the Ryll-Nardzewski Theorem, the closed, oligomorphic subgroups of are precisely the automorphism groups of -categorical structures with domain .

Recall that a topological group is extremely amenable if whenever is a -flow, that is, a non-empty, compact Hausdorff -space on which acts continuously, then there is a -fixed point in . The starting point for this paper is the following question, raised in [7] and [29, Question 1.1].

Question 1.1.

Suppose is a closed, oligomorphic permutation group on a countable set. Does there exist a closed, extremely amenable, oligomorphic subgroup of ?

The question can be formulated in other ways. For example, it asks, given a countable -categorical structure , does there exist an -categorical expansion of whose automorphism group is extremely amenable? Using the Kechris, Pestov, Todorčević (KPT for short) correspondence from [24], the question can be phrased in terms of Ramsey classes and Ramsey lifts, and in this form, it was asked by the third author as a question about the characterisation of Ramsey classes [31]. More precisely, suppose is a first-order language and is a Fraïssé class of finite -structures. Thus there is a countable homogeneous -structure with as its class of (isomorphism types of) finite substructures. Suppose is a language extending . We say that a class of finite structures is an expansion (lift in [20]) of if the -reducts (-shadow in [20]) of the structures in form the class ; it is a Ramsey expansion (or Ramsey lift) if additionally it is a Ramsey class. The above question is then asking, in the case where has only finitely many isomorphism types of structure of each finite size, whether there is a Ramsey expansion of with the same property.

We discuss below some of the motivation for this question, but first we state some of our main results, showing that Question 1.1 has a negative answer in general.

Theorem 1.2.

There exists a countable, -categorical structure with the property that if is extremely amenable, then has infinitely many orbits on . In particular, there is no -categorical expansion of whose automorphism group is extremely amenable.

Using [40], or [4], we then have the following corollary, answering Question 1.5 in [29]:

Corollary 1.3.

There is a closed, oligomorphic permutation group on a countable set whose universal minimal flow is not metrizable.

As a direct corollary to the results in Section 5, we also show:

Corollary 1.4.

There is a closed, oligomorphic permutation group which has a metrizable minimal flow where all -orbits are meagre.

The example which gives these results is Hrushovski’s construction of an -categorical pseudoplane from [18]. This is one of a variety of constructions of countable structures which we shall refer to as Hrushovski constructions. Details will be given later in this paper (in Section 4; also Section 3.4). One feature that all of the variations on the construction have is that interprets a sparse graph . In this case, has a continuous action on the compact space of all orientations of this graph (see Definition 3.5). This is the main tool and object of study in this paper. In Section 3, we describe and use it to prove Theorem 1.2 (in the more general form of Theorem 3.7). As an additional benefit, we also use it (in Section 3.3) to give a simple proof of a general result (Theorem 3.8) about non-amenability of which generalises results in [15]. The argument we use shows that in Theorem 1.2 we may take also having the property that there is no -categorical expansion of with amenable automorphism group (Corollary 3.11). In Theorem 5.2, we give examples where every minimal subflow of has all orbits meagre, thereby proving Corollary 1.4.

The results of Kechris, Pestov and Todorčević in [24] make a strong connection between the study of continuous actions of automorphism groups of countable structures on compact spaces (‘topological dynamics’) and Ramsey properties of classes of finite structures (‘structural Ramsey theory’). In particular, if preserves a linear ordering on and the language for is such that two tuples are in the same -orbit iff they have the same quantifier free type (that is, is homogeneous), then is extremely amenable if and only if the class of finite substructures of is a Ramsey class.

More generally, say that a subgroup is a co-precompact subgroup of if, for every , every -orbit on is a union of finitely many -orbits. If is closed, co-precompact in and extremely amenable, then ([24, 37]) the completion of the quotient space with respect to the right uniformity on is compact and the universal minimal flow of is isomorphic to a minimal subflow of this. Thus, if one has a co-precompact, extremely amenable subgroup of , then one has control over the universal minimal flow of . Question 1.1 asks whether one is guaranteed such a subgroup in the case where is -categorical.

The above analysis shows that if has a co-precompact extremely amenable subgroup, then its universal minimal flow is metrizable. In fact, the converse is also true: if is metrizable, then there is a comeagre -orbit on and the stabilizer of a point in this orbit is extremely amenable and co-precompact in . This is proved by Zucker in [40] and, independently, by Ben Yaacov, Melleray and Tsankov in [4]. (The latter builds on work in [29] and proves the result for arbitrary Polish groups .)

Most proofs that a particular subgroup is extremely amenable make use of structural Ramsey theory. One identifies as the automorphism group of a homogeneous structure and shows that is a Ramsey class (of ordered structures). Many examples of this can be found in the paper [20] and in the surveys [5, 35].

The question of finding extremely amenable subgroups of , or equivalently, finding good Ramsey expansions of , also has applications in the study of reducts of (see [6]

) and hence to classifying constraint satisfaction problems with template

. The paper [23] by Ivanov also mentions the question of whether, if is amenable, then it has a precompact extremely amenable subgroup.

Our results show that the general problem of describing the universal minimal flow (and hence, all minimal flows) of a closed subgroup of is more complicated than the above picture suggests, even in the case where the subgroup is the automorphism group of an -categorical structure (and therefore Roelcke precompact). In our examples, does not necessarily have the co-precompact extremely amenable subgroup needed to make the machinery work. Moreover, we show in Section 5 that for our examples, again in contrast to the above picture, minimal subflows of the space of orientations have all orbits meagre. So there are metrizable -flows which have no comeagre orbit.

It should be noted that Question 1.1 remains open for where is a structure which is homogeneous for a finite relational language (the Hrushovski constructions require an infinite language for homogeneity).

In the Section 6 we prove some positive results for our examples. We study two versions of the Hrushovski construction. The more technically challenging of these is the -categorical case considered in Section 6. Whilst this is perhaps the more important case from the point of view of Question 1.1, the other case is natural and of interest in its own right. In each case, we have an amalgamation class of finite sparse graphs and a distinguished notion of ‘strong substructure’. There is an associated Fraïssé limit and, in each case, for we identify a maximal extremely amenable closed subgroup of , corresponding to an ‘optimal’ Ramsey expansion of (Theorem 6.9).

These positive results raise the possibility that there might still be a weaker statement than the KPT correspondence in [37] which holds more generally. But in any case, the whole KPT-type correspondence for expansions is more complicated than perhaps was thought. The Hrushovski construction leads to interpreting classes of structures which display a complicated behaviour and interplay of related notions: Ramsey classes, the Expansion Property (Definition 2.17) and EPPA (Definition 2.25).

Acknowledgements: The first author thanks Todor Tsankov for numerous helpful discussions about the material in this paper, particularly about the proof of Theorem 3.8.

2. Background

For the convenience of the reader, we provide some background material on homogeneous structures, automorphism groups and Ramsey classes. Although we work with more general classes of finite structures than the Fraïssé classes in, for example, [24, 37], there is little that is new here and the reader who is familiar with this type of material can proceed to the following sections, referring back to this section where necessary.

We briefly review some standard model-theoretic notions. Let be a first-order relational language involving relational symbols each having associated arities denoted by . An -structure is a structure with domain , and relations , . The elements of the domain will often be referred to as vertices of the structure.

The language is usually fixed and understood from the context (and it is in most cases denoted by ). If the set is finite we call a finite structure. We consider only structures with finitely or countably infinitely many vertices.

A homomorphism is a mapping such that for every we have . If is injective, then is called a monomorphism. A monomorphism is an embedding if the implication above is an equivalence. If is an embedding which is an inclusion then is a substructure of . For an embedding we say that is isomorphic to and is also called a copy of in .

Finite structures will often be denoted by capital letters such as , , etc. and infinite structures by . We use the same notation for a structure and its domain and all substructures considered will be full induced substructures. The automorphism group of a structure is denoted by .

2.1. Fraïssé classes

Suppose is a first-order relational language. A countable -structure is called (ultra)homogeneous if isomorphisms between finite substructures extend to automorphisms of . If is -saturated, this is equivalent to the theory of having quantifier elimination. A homogeneous structure is characterised by its age, the class of isomorphism types of its finite substructures. This class satisfies the hereditary, joint embedding and (using the homogeneity) amalgamation properties. Conversely, if is a class of countably many isomorphism types of finite -structures which has these properties, then there is a countable homogeneous structure whose age is . All of this is the classical Fraïssé theory, initiated in [13].

In one direction, this result can be seen as a method for constructing homogeneous structures from amalgamation classes of finite structures. There are several generalisations of this method and we shall state one of these, mostly following the presentation of Section 3 of [3] and the notes [9]. Essentially, we take Fraïssé’s original construction, but instead of working with all substructures (equivalently, all embeddings between structures) we work with certain distinguished substructures, which we will call strong substructures. Embeddings with strong substructures as their image will be called strong embeddings. Other generalisations are possible (though the basic structure of the proof is always the same). For example, general category-theoretic versions of the Fraïssé construction can be found in [8] and [26].

Definition 2.1.

Suppose is a first-order language and is a class of finite -structures, closed under isomorphisms. Suppose is a distinguished class of embeddings between elements of . Assume that is closed under composition and contains all isomorphisms. Furthermore, suppose that has the following property:

  • whenever and is in and is a substructure of which contains the image of , then the map with for all is in .

Then we say that is a strong class and refer to the elements of as strong embeddings. If is a substructure of and the inclusion map is in , then we say that is a strong substructure of and write . Thus an embedding between structures in is in if and only if . Henceforth, we suppress the notation and refer to the strong class as . We sometimes refer to the strong embeddings of this class as -embeddings. In this notation the condition (*) says that if satisfy and , then .

If consists of all embeddings between structures in , then we write for the class, instead of .

Note that if are in a strong class , then and implies that .

Definition 2.2.

Suppose is a strong class of finite -structures and the -structure is the union of a chain of finite substructures . If is a finite substructure of , we write to mean that for some , and say that is a strong substructure of .

Remark 2.3.

It is important to note that the above definition does not depend on the choice of the sequence of . Indeed, suppose also that is the union of the finite substructures . Suppose . There exist with ; as , the condition (*) implies that and so . Note that this also shows that if then if and only if .

We also note that when we come to consider specific examples of strong classes, the definition of the strong embeddings will extend naturally to maps between arbitrary structures. We will usually omit the verification that the extension agrees with that in the above definition.

Definition 2.4.

Suppose is a strong class of finite -structures. An increasing chain

of structures in is called a rich sequence if:

  1. for all there is some and a strong embedding ;

  2. for all strong there is and a strong such that for all .

A Fraïssé limit of is an -structure which is the union of a rich sequence of substructures.

Definition 2.5.

We say that the strong class has the amalgamation property (for strong embeddings), AP for short, if whenever are in and and are strong, there is and strong (for ) with . The class has the joint embedding property, JEP for short, if for all there is some and strong and .

If and for all , then the JEP is a special case of the AP.

Theorem 2.6.

Suppose is a strong class of -structures which satisfies:

  1. There are countably many isomorphism types of structures in .

  2. The class has the Joint Embedding and Amalgamation Properties with respect to strong embeddings.

Then rich sequences exist and all Fraïssé limits are isomorphic. Moreover, if is a Fraïssé limit and is an isomorphism between strong finite substructures of , then extends to an automorphism of .

We refer to the last property in the above as -homogeneity (or strong-map homogeneity) and say that the strong class is an amalgamation class (for strong maps) if conditions (1, 2) hold. Note that in the case where is closed under substructures and consists of all embeddings between structures in , this result is the classical Fraïssé Theorem.

Remarks 2.7.

Many of the examples of amalgamation classes of relational structures in this paper will be free amalgamation classes. If are structures in , then by the free amalgam of and over we mean the structure whose domain is the disjoint union of and over and in which the relations (for in the language) are just the unions of the the relations on and . If is always in and , then we say that is a free amalgamation class.

Remarks 2.8.

Suppose in Theorem 2.6 has only finitely many isomorphism types of structure of each finite size. Suppose also that there is a function such that if and with , then there is with and . Then the Fraïssé limit is -categorical.

To see this we note that has finitely many orbits on . Indeed, by -homogeneity there are finitely many orbits on and any can be extended to an element of this set.

Much of this paper will be concerned with expansions of Fraïssé limits, or their corresponding amalgamation classes. The following notions will be useful.

Definition 2.9.

Suppose that are first-order languages and , are strong classes of finite - and -structures respectively. We say that is a strong expansion of if is the class of -reducts of the structures in and:

  1. whenever is a strong map in , the map between the -reducts is strong in ;

  2. if is a strong map in and is an expansion of , then there is an expansion of such that is strong in .

Theorem 2.10.

Let be first-order languages. Suppose that is an amalgamation class of finite -structures which is a strong expansion of the strong class of -structures. Then is an amalgamation class. Moreover, if , denote the Fraïssé limits of and respectively, then the -reduct of is isomorphic to .

Proof.

Suppose are strong embeddings in , for . We can expand to structures in so that is strong. We can then expand to a structure in so that is strong. The amalgamation property in gives that there exist and strong with . Passing to the -reducts, we obtain the amalgamation property for . So this is an amalgamation class.

Similarly, suppose is a rich sequence for . Then the sequence of -reducts is easily seen to be rich for . The result follows. ∎

Examples of such strong expansions will be found in later sections (for instance, see Section 3.4).

2.2. Ramsey classes

Throughout this subsection, will be a first-order language and we work with strong classes of finite -structures as in Section 2.1. We shall say what it means for to be a Ramsey class and in the next subsection,we state the KPT correspondence and associated results in this context. Similar statements (about special class of maps) can be found in the paper of Zucker [40] and in [15]. In the case where is closed under substructures and is just the usual notion of substructure, this is just the usual notion of Ramsey class and the KPT correspondence. The statements which we give can all be deduced from this case either by simple adaptations of the proofs, or, more directly, by expanding the language in a suitable way. Indeed, the latter is the approach taken by the second and third authors in [20], where the results are stated for classes of structures with closures. This latter paper is the main source of the Ramsey results which we will use in the later sections, so we will state its results in detail.

Suppose is a first-order language and that is a strong class of finite -structures. For we denote by the set of all strong substructures of which are isomorphic to . Note that by the condition (*) in Definition 2.1, if and , then , that is, the strong copies of in are precisely the strong copies of in whose domains are subsets of .

We say that is a Ramsey class if it is a strong class and for all and all , there is a structure in such that the following Ramsey property holds: whenever is partitioned into classes (‘colours’), there is such that the elements of all lie in the same class (that is, they have the same colour). In this case, we write:

Note that here we are restricting to strong substructures throughout (without incorporating this into the notation) and it is of course enough to consider this in the case .

More generally, we say that has finite Ramsey degree if there is a natural number such that for all with and all , there is in such that whenever is coloured with colours, there is such that is coloured with at most colours. The least such is then the Ramsey degree of in . Note that if this is equal to 1 for all , then has the Ramsey property.

The Ramsey property is sometimes defined with respect to colourings of embeddings from to and . If the structures in are rigid (that is, have trivial automorphism group), then there is no difference between these notions. This is the case if, for example, each structure in has a linear ordering as part of the structure.

In the case where all substructures are strong, it is a well known observation of the third author (cf. [36], for example) that (under mild extra conditions) Ramsey classes are amalgamation classes. We note that the usual argument also applies in our current context (of strong maps).

Theorem 2.11.

Suppose that is a first-order language and is a Ramsey class of finite, rigid -structures with the joint embedding property. Then has the amalgamation property.

Proof.

Let be strong embeddings forming our amalgamation problem. As the structures are rigid, it is enough to find which contains strong copies of having a copy of as a common strong substructure.

There is some which contains strong copies of and (using JEP). Find with . Colour the elements of according to whether or not they are contained in a strong copy of in . There is a monochrome copy of . As it contains a strong copy of , all the strong copies of in it are in a strong copy of in . But this includes the which is in the copy of in . ∎

We now state, using this terminology, the general Ramsey result which we need. While the original Fraïssé construction (where all embeddings are strong) generalises naturally to strong amalgamation classes, it is the essence of our examples to show that this is not the case for the construction of Ramsey objects. For this reason, the papers [20, 12] use an alternative approach, representing strong substructures by means of functions which are part of the structures themselves instead of by an external family of strong embeddings. We review the main terminology and results of [12] (using several results of [20]) which will be needed here.

First we introduce a notion of structure involving functions in addition to relational symbols. Unlike the usual model-theoretic functions, functions used here are partial, multi-valued and symmetric. Partial functions allow the easy definition of free amalgamation classes and the symmetry makes it possible to explicitly represent strong embeddings within the structure itself, while keeping all automorphisms of the original structure.

Let be a language involving relational symbols and function symbols each having associated arities denoted by for relations and domain arity, , range arity, , for functions. Denote by the set of all subsets of consisting of elements. An -structure is a structure with domain , functions , , and relations , . The set is called the domain of the function in .

Given two -structures and , we say that is a substructure of and write if the following holds:

  1. the domain of is a subset of domain of ,

  2. for every relation it holds that is the restriction of to , and,

  3. for every function it holds that is the restriction of to and moreover for every it holds that .

Embeddings are defined analogously (a substructure then expresses the fact that inclusion is an embedding).

If are -structures and are embeddings, then an -structure together with embeddings is called an amalgamation of and over if for all . It is a free amalgamation of and over if only if and moreover there are no tuples in any relations of and no tuples in (with and ) using vertices of both and , and .

In the case where consists only of relation symbols, note that this coincides with the usual notion of free amalgamation (as in Remarks 2.7).

Suppose now that is a strong class of -structures. Suppose, moreover, that strong substructures are closed under intersections (and therefore there is an associated notion of closure). Then there is a standard way to turn this class into amalgamation class which is closed for substructures: We can expand to a language by adding partial functions , for every , from -tuples to subsets of size . On a structure , map every set of elements to the smallest strong substructure of containing , that is and leave the functions undefined otherwise. Note that doing this does not affect the automorphisms of . The resulting class is then a strong class (where strong maps are all embeddings) and if is an amalgamation class, then so is .

Observe that even if is a free amalgamation class, then constructed in this standard way is not necessarily a free amalgamation class. However in cases discussed here we will be able to omit some of the functions to obtain which is closed for free amalgamation. This will allow us, in Section 6 to apply the following theorem to show that such a class has an easy Ramsey expansion.

Given a class of -structures, denote by the class of all structures where and is a linear ordering of the domain of . The following is combination of Theorems 1.3 and 1.4 of [12] which will be applied in Section 6.3. The Expansion Property is defined in Definition 2.17 below.

Theorem 2.12.

Let be a language (involving relational symbols and partial functions) and let be a free amalgamation class. Then is a Ramsey class and moreover there exists a Ramsey class such that is a strong expansion of having the Expansion Property with respect to .

2.3. The KPT correspondence

The fundamental connection between Ramsey classes and topological dynamics is the following result of Kechris, Pestov and Todorčević which we formulate in the following way in the context of strong maps. Recall that an amalgamation class is a strong class satisfying the conditions (1,2) of Theorem 2.6.

Theorem 2.13 ([24], Theorem 4.8; [37], Theorem 1).

Let be a first-order language and an amalgamation class of finite, rigid -structures. Let be the Fraïssé limit of the class. Then is extremely amenable if and only if is a Ramsey class.

Now we modify this for expansions.

Definition 2.14.

Suppose that are first-order languages and consists of relation symbols. Let be an amalgamation class of finite -structures with Fraïssé limit . Suppose that is a class of finite -structures with the properties:

  1. the class of -reducts of is ;

  2. each structure in has finitely many isomorphism types of expansions in ;

  3. if and is a substructre of with (that is, the corresponding -reducts satisfy ), then ;

  4. if is strong in and is an expansion of , then there is an expansion of such that is an embedding.

Then we say that as a reasonable class of expansions of .

The above terminology follows [40]. Note however that we include (ii) as part of the definition rather than referring to it as ‘precompactness’. We remark that if is a strong expansion of as in Definition 2.9, then satisfies conditions (i), (iii) and (iv) in the above. To see that (iii) holds, suppose and . Let be the -reducts of . So and by (ii) in Definition 2.9, there is an expansion of in which the induced substructure on is and . It follows that as is a strong class.

Suppose that is a reasonable class of expansions of as in the above definition and is the Fraïssé limit of . We shall consider the set of -expansions of which have the property that for every finite , the -structure induced on in is in the class . This is a topological space where a basic open set is given by considering the expansions in which is a fixed structure in (for a finite ). The property (ii) in the definition implies that is compact and it follows from properties (iii) and (iv) that is non-empty. Note that if consists of finitely many relation symbols of arities (for ), then can be identified with a subset of with the product topology. In general, embeds in an inverse limit of such spaces, by the property (ii) in Definition 2.14.

With this notation, we have the following, summarising the above and statements in [24, 37]:

Theorem 2.15.

Suppose is a reasonable class of -expansions of the amalgamation class of finite -structures. Let denote the Fraïssé limit of . Then the space is a non-empty, compact space on which acts continuously.

Lemma 2.16.

Let be an amalgamation class with Fraïssé limit and . Let be a reasonable class of expansions of and suppose is a subflow of the -flow . Then there is which is a reasonable class of expansions of such that .

Proof.

Let consist of isomorphism types of structures induced on by expansions in , for all finite . Then clearly satisfies properties (i), (ii) in the definition of reasonableness. Property (iii) follows from the -homogeneity of .

Clearly we have . We claim that . Let be an expansion of in . We show that is in the closure in of and this will be enough. Suppose is finite. There is and and an isomorphism from (the induced structure on in ) to . By -homogeneity of , there is which extends this map. By considering we obtain with . This gives what we need. ∎

The following (from [37], Theorem 4; see also Proposition 5.5 in [40]) gives a criterion for minimality of the -flow . It relates to the notion of Expansion Property defined as follows:

Definition 2.17.

Let be a reasonable class of expansions of the amalgamation class . We say that has the Expansion Property (EP for short, or Lift Property in [20]) with respect to if, for every there is in with the property that for any expansions of in , there is an embedding which is -strong. (If the extra structure imposed by is a total order, this is usually called the Ordering Property (cf. [35]).)

Theorem 2.18.

With the above notation, suppose that is a reasonable class of expansions of the amalgamation class and is the Fraïssé limit of . Then the -flow is minimal if and only if has the Expansion Property with respect to .

Remarks 2.19.

Note that as every -flow has a minimal subflow, it follows from the above that if is a reasonable class of expansions of the amalgamation class , then there is a reasonable sub-class which has the Expansion Property with respect to .

Suppose, as in Theorem 2.10, that are first-order languages with relational. Suppose is an amalgamation class of finite -structures which is a strong expansion of the strong class (cf. Definition 2.9). The latter is also an amalgamation class, by Theorem 2.10 and its Fraïssé limit is the -reduct of the Fraïssé limit of . Thus is a closed subgroup of . We will say that is a precompact strong expansion of when is a co-precompact subgroup of . This means that every -orbit on (for ) splits into finitely many -orbits. This is a stronger condition that property (ii) in Definition 2.14 and so, by previous remarks, is a reasonable class of expansions of and we can consider the -flow .

By the homogeneity of the structures , co-precompactness of in translates into the following condition on the classes:

Precompactness: If , there are finitely many and strong -embeddings (preserving the -structure) such that whenever and is an -embedding (preserving the -structure), then there is and a -embedding such that .

We then have the following version of [24], Theorem 10.8 and [37], Theorem 5. See also [40], Theorem 5.7.

Theorem 2.20.

Let be first-order languages with relational. Suppose is a strong amalgamation class of finite -structures with Fraïssé limit . Suppose that is a precompact strong expansion of consisting of rigid -structures. If is a Ramsey class and has the Expansion Property with respect to , then the -flow is the universal minimal flow for . It has a comeagre orbit consisting of expansions of isomorphic to the Fraïssé limit of .

We note a group-theoretic consequence of the above. Suppose is a closed permutation group on a countable set. We can regard as the automorphism group of some homogeneous structure . Suppose is a closed, extremely amenable subgroup of . We can consider this as the automorphism group of a homogeneous expansion of . If this is a precompact expansion and has the Expansion Property with respect to (abusing terminology, we will say that has EP as a subgroup of ), then the above result gives a description of , the universal minimal flow of . Consider the quotient space with the quotient of the right uniformity on and denote by the completion. By precompactness, this is compact, metrizable and embeds homeomorphically as a comeagre set. It is a -flow which, by the EP, is minimal. Extreme amenability of then implies that is isomorphic to as a -flow. This then yields the following result.

Theorem 2.21.

Suppose is a closed permutation group on a countable set and are closed, extremely amenable, co-precompact subgroups of with EP. Then are conjugate in . Moreover, is maximal amongst extremely amenable subgroups of .

Proof.

The universal minimal flow is isomorphic to and is a comeagre -orbit in this completion.

So there is homeomorphism which is a -morphism. This must map the comeagre orbit to the comeagre orbit, so maps the coset to some coset . The stabilisers of these points must be identical, so , as required.

For the maximality part, we first observe that if , then . By precompactness each -orbit on -tuples splits into a finite number of -orbits, and the same number of -orbits. It follows that have the same orbits on -tuples and so are equal. Now suppose that and is closed and extremely amenable. Clearly is co-precompact in and the -map extends to a continuous surjection . It follows that is a minimal -flow and so has EP. From the above, we obtain as required. ∎

Remarks 2.22.

By Theorem 6 of [37], we know that if a closed permutation group on a countable set has a closed, co-precompact, extremely amenable subgroup, then it has one satisfying EP. By the above, the latter is maximal amongst extremely amenable subgroups. In this case, is it also true that every maximal co-precompact, closed, extremely amenable subgroup of satisfies EP? In other words, are the maximal, co-precompact, closed extremely amenable subgroups of all conjugate in ?

2.4. Comeagre orbits and the weak amalgamation property

Suppose that is an amalgamation class of -structures and is a reasonable class of expansions of . Then is still a strong class, but of course it need not be an amalgamation class. Following [25] we say that has the weak amalgamation property if for all , there is and a strong map such that for all strong maps (for ), there exist and strong maps with for all . A similar property (the almost amalgamation property) is introduced by Ivanov in [22]. We then have:

Lemma 2.23.

Suppose that is a reasonable class of expansions of the amalgamation class . Let be the Fraïssé limit of , let and consider the -flow . If does not have the weak amalgamation property, then all -orbits on are meagre.

Proof.

Suppose that witnesses that does not have the weak amalgamation property. Let (so we think of this as the ‘extra structure’ on for a particular expansion in ) and let denote the pointwise stabiliser in of . We claim that the -orbit containing is nowhere-dense in . As is of countable index in , it then follows that the -orbit is a meagre subset of .

So suppose for a contradiction that is dense in the open set . We may assume that there is a finite with and such that . (Again, by the notation we mean the structure together with the additional structure it has as a member of .) Clearly we have that , the induced structure on in , is equal to .

Note that, by reasonableness of , if , then there is and a -embedding which is the identity on . As is dense in , there is such that . Then is a -embedding which is the identity on . It follows that has the weak amalgamation property over : a contradiction. ∎

Remarks 2.24.

Arguments in [25, 22] show that, with the above notation, if has the joint embedding property and the weak amalgamation property, then there is a comeagre -orbit on .

2.5. EPPA and amenability

The following is a modification of a well-known definition.

Definition 2.25.

Suppose is a strong class of finite structures (as in Section 2.1). A strong partial automorphism of is an isomorphism for some . We say that has the extension property for strong partial automorphisms (sometimes called the Hrushovski extension property or EPPA) if whenever there is with and such that every strong partial automorphism of extends to an automorphism of .

Recall that a topological group is amenable if, whenever is a

-flow, then there is a Borel probability measure

on which is invariant under the action of . Thus, if is a Borel set and , then . Of course, if is extremely amenable, then it is amenable (if is fixed by , then take for the probability measure which concentrates on ).

The following is due to Kechris and Rosendal ([25], Proposition 6.4). The terminology is as in Section 2.1 here.

Theorem 2.26.

Suppose is an amalgamation class of finite structures with Fraïssé limit . Let . Suppose has the extension property for strong partial automorphisms (EPPA). Then:

  1. There exist compact subgroups of such that is dense in .

  2. is amenable.

Note that (ii) follows from (i) by standard results on amenability.

3. -Sparse graphs and their orientations

3.1. The space of orientations

In this section, the structures we work with are graphs and directed graphs (digraphs), so we use notation which is closer to standard graph-theoretic notation. Note that our directed graphs will be asymmetric: we do not allow loops nor vertices where both and are directed edges.

An undirected graph will be denoted as , ; if then ; so is the induced subgraph of on .

A digraph will be denoted as , where .

Definition 3.1.

Let . We say that a graph is -sparse if for all finite we have . An infinite graph is sparse if it is -sparse for some .

Note that this differs from the use of “sparse” in, for example, [32].

Remark 3.2.

We could consider the more general notion of a sparse relational structure. For example, if and is an -ary relation on , then we say that is -sparse if for all we have . However, we can then consider the graph which has edges where for some . This graph is -sparse. Thus, the results below apply more generally to sparse relations. Of course, dealing with graphs simplifies the reasoning.

Definition 3.3.

Let . A graph is -orientable if there is a digraph in which the out-valency of each vertex of is at most and such that for all ,

In this case, we refer to (or ) as a -orientation of .

So, informally, a -orientation of is obtained by choosing a direction on each edge of in such a way that no vertex has more than directed edges coming out of it. Note that if a graph is -orientable, then its edge-set can be decomposed into subsets such that each graph is -orientable. Moreover, a graph is -orientable if and only if each of its connected components is a ‘near-tree’: a tree with at most one extra edge. Thus, -orientability is closely related to -arboricity. The following is well-known to graph-theorists [30], but we include a proof.

Theorem 3.4.

A graph is -orientable if and only if it is -sparse.

Proof.

If is -orientable and is a finite subset of , then then the number of edges in is (at most) the number of directed edges in the induced sub-digraph on in a -orientation of . Clearly this is at most .

Note that by a compactness (or König’s Lemma) argument, it suffices to prove the converse in the case where is finite, so we now assume this. For a -orientation of we need to choose, for each edge one of the vertices to be the initial vertex of the directed edge. We need to do this so that the resulting digraph has out-valency at most .

Consider copies of the vertex set (where ) and form a bipartite graph with parts and . We have an edge (where and , ) in this bipartite graph if and only if . We show that the condition of Hall’s Marriage Theorem holds and hence there is a matching of into . Indeed, if , let be the union of the edges in . Then the number of vertices in adjacent to is and as required.

Fix a matching in . We orient an edge of by taking the directed edge precisely when is matched with some under the matching. This is a -orientation. ∎

We use the following special case of the construction in Theorem 2.15.

Definition 3.5.

Suppose that is a -sparse graph. We let

be the set of -orientations of . Identifying

with its characteristic function, we can view

as a subset of . We give the latter the product topology (where has the discrete topology). We give the subspace topology and refer to it as the space of -orientations of . Note that the automorphism group acts continuously on via its diagonal action on and is invariant under this action.

Of course, this depends on the particular , but we omit this dependence in the notation.

Lemma 3.6.

Suppose is a -sparse graph. Then is an -flow.

Proof.

By Theorem 3.4, is non-empty. We know that is an invariant subspace of the -flow , so it suffices to observe that it is a closed subspace. But if is not a -orientation of , then this is witnessed on some finite subset of . So if agrees with on