New Ramsey Classes from Old

Let C_1 and C_2 be strong amalgamation classes of finite structures, with disjoint finite signatures sigma and tau. Then C_1 wedge C_2 denotes the class of all finite (sigma cup tau)-structures whose sigma-reduct is from C_1 and whose tau-reduct is from C_2. We prove that when C_1 and C_2 are Ramsey, then C_1 wedge C_2 is also Ramsey. We also discuss variations of this statement, and give several examples of new Ramsey classes derived from those general results.


Introduction
A class of relational structures is a Ramsey class if it satisfies a strong combinatorial property that resembles the statement of Ramsey's theorem. Surprisingly many classical classes of relational structures turned out to be Ramsey classes. Nešetřil [12] asked whether one may classify all Ramsey classes that are closed under induced substructures and have the joint embedding property, and he indicated a link to model-theoretic classification of countably infinite homogeneous structures as an approach to such a classification. This program has recently attracted attention because of a fascinating correspondence between Ramsey classes and the concept of extreme amenability in topological dynamics [10]. We would also like to mention that Ramsey classes play an important role in classifications of first-order reducts of homogeneous relational structures [5], and for complexity classification of infinite-domain constraint satisfaction [2]. Establishing that a class has the Ramsey property is often a substantial combinatorial challenge, and we are therefore interested in general transfer principles that allow to prove the Ramsey property by reducing to known Ramsey classes; this will be the topic of this text.
For structures A and B over the same relational signature, let B A denote the set of all embeddings of A into B. When f is such an embedding, we write f [A] for the copy of A in B that is induced by the image of A under f in B. The partition arrow C → (B) A r means that for every function χ : C A → [r] (a colouring with r colours) there exists g ∈ C B such that χ is constant on g [B] A . In this case we call g[B] a monochromatic copy of B in C. A class of finite relational structures C has the Ramsey property (with respect to embeddings) 1 if for all A, B ∈ C and r ∈ N there exists a C ∈ C such that C → (B) A r . It is easy to see that every class C with the Ramsey property only contains rigid structures, that is, structures with only one automorphism, the identity. Note that an ordered structure, that is, a structure that has a strict linear order as one of its relations, is always rigid. A class of relational structures that is closed under isomorphisms and has the Ramsey property is also called a Ramsey class.
Examples of Ramsey classes are • LO, the class of all finite linear orders (this is equivalent to Ramsey's original theorem); • the class of all ordered finite graphs (see [13]); • the class of all ordered K n -free graphs (see [13]); • the class of all finite partially ordered sets with a linear extension (see [12]; • the class of all finite tournaments with an additional linear orde; • the class of all finite naturally ordered C-relations on a finite set (this is essentially due to [11]; see [4]).
It is of major interest in combinatorics to obtain a more systematic understanding of the question which classes of structures have the Ramsey property. Nešetřil made the important observation that Ramsey classes that are closed under taking induced substructures are linked with the concept of amalgamation in model theory. We say that a class of structures has the amalgamation property if for all A, B 1 , B 2 ∈ C and embeddings e 1 : A → B 1 and e 2 : A → B 2 there exists a C ∈ C and embeddings f i of B i to C such that f 1 (e 1 (a)) = f 2 (e 2 (a)) for all a ∈ A. We call (A, B 1 , B 2 , e 1 , e 2 ) the amalgamation diagram, and (C, f 1 , f 2 ) an amalgam of the diagram (A, B 1 , B 2 , e 1 , e 2 ) (in C). If C has the amalgamation property only for the special case that A is empty, we say that C has the joint embedding property (here, our assumption that the signature is relational becomes important). The mentioned link between Ramsey theory and amalgamation is that every class C of rigid finite relational structures that is closed under isomorphisms and induced substructures, and that has the joint embedding and the Ramsey property also has amalgamation property [12]. Classes of finite structures with countably many non-isomorphic structures that are closed under isomorphisms, induced substructures, and have the amalgamation property are called amalgamation classes.
The age of a relational structure Γ is the class of all finite structures that embed into Γ. A structure is homogeneous if any isomorphism between finite induced substructures of Γ can be extended to an automorphism of Γ. When C is an amalgamation class, then Fraïssé's theorem shows that there exists a countably infinite homogeneous structure Γ whose age is C (see e.g. [9]). The structure Γ is unique up to isomorphism, and called the Fraïssé-limit of C; these homogeneous limit structures will play an important role in the proof of our main result. The significance of Nešetřil's observation is that the transition to countable homogeneous structures brings new tools for the systematic understanding of Ramsey classes; and indeed, under some additional assumptions, there are many classification results for homogeneous structures (such as the classification of all homogeneous directed graphs [8]).
A strong amalgam of an amalgamation diagram (A, B 1 , B 2 , e 1 , e 2 ) is an amalgam . A class C has strong amalgamation if every amalgamation diagram has a strong amalgam in C. We say that C is a strong amalgamation class if C is closed under isomorphisms, induced substructures, and has the strong amalgamation property. An example of a strong amalgamation class is LO. Homogeneous structures Γ that arise as the Fraïssé-limits of strong amalgamation classes can be characterized via algebraic closure: in this context, we define the algebraic closure acl(A) of a finite subset A = {a 1 , . . . , a n } of the domain of Γ to be the set of all those elements of Γ which lie in finite orbits of the expansion (Γ, a 1 , . . . , a n ) of Γ by the constants a 1 , . . . , a n . Proposition 1.1 (see (2.15) in [7]). The age of a homogeneous structure Γ has strong amalgamation if and only if for any finite subset A of the domain of Γ, acl(A) = A. Definition 1.2 (from [7]). Let C 1 and C 2 be strong amalgamation classes with disjoint signatures σ and τ . Then C 1 ∧ C 2 denotes the class of all finite (σ ∪ τ )-structures whose σ-reduct is from C 1 and whose τ -reduct is from C 2 .
It is clear that C 1 ∧ C 2 also has strong amalgamation. In Section 3, we prove the following. Theorem 1.3. Let C 1 and C 2 be strong amalgamation classes with the Ramsey property, with disjoint finite signatures σ and τ . Then C 1 ∧ C 2 is Ramsey.
The following is an immediate consequence of Theorem 1.3 and the previously known Ramsey results mentioned above.

The class of all all naturally ordered C-relations on finite sets that additionally carry a poset and a linear extension of this poset.
Item 1. in Corollary 1.4 has been obtained independently by Böttcher and Foniok [6] and by Sokić [14]. It is clear that the list can be prolonged easily.
To prove the statement in item 1., Sokić developed a technique called cross construction; also see [15]. He also proved item 2. and 3. in Corollary 1.4. The present work has been found independently from [15], and it would be interesting to compare our approach with the approach in [15].
Homogeneous structures with a finite relational signature are ω-categorical, that is, their first-order theory has only one countable model up to isomorphism. We can weaken the assumption of having a finite signature slightly, and prove the following stronger version which captures several additional interesting classes (see Corollary 1.8).
Theorem 1.5. Let C 1 and C 2 be strong amalgamation classes with ω-categorical Fraïssélimits. If C 1 and C 2 have disjoint relational signatures and the Ramsey property, then C 1 ∧ C 2 is also Ramsey.
To show the Ramsey property for even more classes, we would also like to be able to generate Ramsey classes that only have one linear order in their signature; this can be accomplished using the following proposition whose proof can be found in Section 4. Proposition 1.6. Let C 1 and LO ∧ C 2 be Ramsey classes with strong amalgamation and ω-categorical Fraïssé-limits, and suppose that C 1 , C 2 , and LO have pairwise disjoint relational signatures. Then C 1 ∧ C 2 has the Ramsey property.  • G for the class of all finite graphs, • F n for the class of all finite K n -free graphs, • T for the class of all linearly ordered finite tournaments, • G for the class of all linearly ordered finite graphs, • F n for the class of all linearly ordered finite K n -free graphs, • P for the class of all linearly extended finite posets, • C for the class of all naturally ordered C-relations on a finite set [4], • V for the class of all finite affine vector spaces V , equipped with a 'natural order' (see [10]); the vector spaces will be represented as relational structures with an infinite signature that contains a relation symbol for every affine equation.
Corollary 1.8. Let C 1 be one of the classes T, G, F n , P, C, V, and let C 2 be one of the classes T, G, F n . Then C 1 ∧ C 2 is Ramsey.

Topological Dynamics
Our combinatorial result translates nicely into a result that shows that certain intersections of extremely amenable groups are again extremely amenable, based on a connection between Ramsey theory and topological dynamics (Theorem 1.9). In fact, our presentation of the proof of Theorem 1.5 makes use of this connection, and so we briefly present it in the following. The property of ω-categoricity of a structure Γ can be characterized in terms of the automorphism group of Γ. A countable structure is ω-categorical if and only if its automorphism group is oligomorphic, that is, has only finitely many orbits of n-tuples, for all n. A topological group G is called extremely amenable if every continuous action of G on a compact Hausdorff space has a fixed point. We say that a homogeneous structure Γ is Ramsey if the class of all finite induced substructures that embed into Γ is a Ramsey class. The following is the central result from [10] (in a slightly more general setting -but we focus on ω-categorical structures here). Theorem 1.9 (of [10]). Let Γ be homogeneous and ω-categorical with domain D. Then the following are equivalent.
• Γ is Ramsey and there is a linear order on D with a quantifier-free first-order definition in Γ.
Some explanations are in place, since Theorem 1.9 is only implicitly in [10]. Recall that we color embeddings, and not induced substructures, so the Ramsey property implies rigidity. The equivalence of rigidity and having an invariant linear order for Ramsey structures is stated in Proposition 4.3 in [10]. Also note that our additional assumption that Γ is ω-categorical implies that when a relation (such as the linear order) is preserved by all automorphisms of Γ, then it has a first-order definition in Γ. This is a well-known consequence of the proof of the theorem of Engeler, Svenonius, and Ryll-Nardzewski; see [9]. Finally, homogeneity of Γ implies that first-order formulas are equivalent to quantifier-free first-order formulas (again, see [9]).

Model-Complete Cores
In our proofs, we make use of the concept of model-complete cores of ω-categorical structures. A structure Γ is called a core if every endomorphism 2 of Γ is an embedding. A first-order theory T is called model-complete if all embeddings between models of T preserve all first-order formulas. An ω-categorical structure Γ has a model-complete theory if and only if all self-embeddings e of Γ are locally generated by the automorphisms of Γ, that is, for every finite tuple t of elements from Γ there exists an automorphism α of Γ such that e(t) = α(t) (see e.g. Theorem 3.6.11 in [2]). In this case, we say that Γ is model-complete. The following has been shown in [1] (also see [3]).
Theorem 2.1. Every ω-categorical structure is homomorphically equivalent 3 to a modelcomplete core ∆, which is unique up to isomorphism, and again ω-categorical or finite. The expansion of ∆ by all existential positive definable relations is homogeneous.
The structure ∆ in Theorem 2.1 will be called the model-complete core of Γ. We need the following observation. Proof. Let h be a homomorphism from Γ to ∆, and let i be a homomorphism from ∆ to Γ. Suppose that f is an isomorphism between two finite substructures A, A ′ of ∆. The restriction of i to A and to A ′ is an isomorphism as well, since otherwise the endomorphism x → h(i(x)) of ∆ would not be an embedding, contradicting the assumption that ∆ is a core. By homogeneity of Γ there exists an automorphism α of Γ that extends the isomorphism i • f • i −1 between i(A) and i(A ′ ). The mapping e : x → h(αi(x)) is an endomorphism of ∆, and therefore an embedding. Since ∆ is a model-complete core, this mapping is locally generated by the automorphisms of ∆, and in particular there exists an automorphism β of ∆ such that β(x) = e(x) = f (x) for all x ∈ A. This proves homogeneity of ∆.
We now prove the following. Theorem 2.3. Let Γ be ω-categorical, homogeneous, and Ramsey, and let ∆ be the model-complete core of Γ. Then ∆ is also Ramsey.
Proof. First note that ∆ is homogeneous, by Proposition 2.2. Let h be a homomorphism from Γ to ∆, and let i be a homomorphism from ∆ to Γ. Write D for the domain of ∆, and τ for the signature of Γ and ∆.
By Theorem 1.9, there is a linear order < on the elements of Γ with a quantifier-free first-order definition ϕ(x, y) in Γ. We claim that ϕ defines a linear order on the elements of ∆. If there were three elements x, y, z such that ϕ(x, y), ϕ(y, z), and ϕ(z, x) in ∆, then ϕ(i(x), i(y)), ϕ(i(y), i(z)), and ϕ(i(z), i(x)) in Γ since the restriction of i to {x, y, z} is an embedding. Similarly one can verify that the relation defined by ϕ in Γ is total and antisymmetric.
Since ∆ is ordered by a quantifier-free formula, we can prove the statement that ∆ is Ramsey by coloring finite induced substructures rather than embeddings. Let P, H be two finite substructures of ∆. Note that i(P ) induces in Γ a copy of P since otherwise the endomorphism x → h(i(x)) of ∆ would not be an embedding. Moreover, for every copy Q of P in Γ we have that h(Q) induces a copy of P in ∆. To see this, let α be the automorphism of Γ that maps i(P ) to Q; such an α exists by homogeneity of Γ. Then e : x → h(αi(x)) must be an embedding, and e(P ) = h(Q) which proves the claim.
Let χ : ∆ P → {0, 1} be arbitrary. We define a map ξ : Γ P → {0, 1} by setting ξ(Q) = χ(h(Q)) for every copy Q of P in Γ. Since Γ is Ramsey, we find a copy L of H in Γ such that ξ is constant on L P . By an argument similar as given above, the restriction h ′ of h to L is an isomorphism, and the image of h ′ induces a copy M of H in ∆. We are left with the task to show that χ is constant on M P . So let P 1 , P 2 be two copies of P in M . Let Q 1 , Q 2 be the pre-images of P 1 and P 2 with respect to the embedding h ′ . Then Q 1 and Q 2 are copies of P in L and therefore χ(Q 1 ) = χ(Q 2 ). It follows that ξ(P 1 ) = ξ(P 2 ).
Suppose now that Γ 1 and Γ 2 are ordered, and let G 1 , G 2 be the automorphism group of Γ 1 and Γ 2 , respectively. Then the product action of the direct product G 1 × G 2 on D 2 equals the automorphism group of Γ 1 ⊠ Γ 2 (using that both Γ 1 and Γ 2 are ordered). Proposition 3.1. Let Γ 1 and Γ 2 be ordered homogeneous structures with the same domain D and disjoint signatures. Then Γ := Γ 1 ⊠ Γ 2 is homogeneous as well.
The following is also known under the name product Ramsey theorem. Proposition 3.2. Let Γ 1 and Γ 2 be ω-categorical structures with the same domain D and disjoint signatures. When Aut(Γ 1 ) and Aut(Γ 2 ) are extremely amenable, then the automorphism group of Γ := Γ 1 ⊠ Γ 2 is oligomorphic and extremely amenable.
Proof. It is easy to bound the number of orbits of n-tuples in Γ by the number of orbits of n-tuples of Γ 1 and Γ 2 , so Γ can be seen to be ω-categorical. When G 1 and G 2 are extremely amenable groups, then G 1 × G 2 is extremely amenable as well (see [10]). The statement follows from the observation that Aut(Γ 1 ⊠ Γ 2 ) is the product action of Aut(Γ 1 ) × Aut(Γ 2 ) on D 2 (here we use that Γ 1 and Γ 2 are ordered; see [2] for a more detailed discussion of full products).
Strong amalgamation will be used via the following lemma. A relation is called injective if it only contains tuples with pairwise distinct entries. Lemma 3.3. Let τ be a relational signature, and let Γ be a homogeneous τ -structure such that the class of all finite τ -structures that embed into Γ has the strong amalgamation property. Suppose moreover that all relations of Γ are injective. Then every finite structure F that homomorphically maps to Γ also has an injective homomorphism to Γ.
Proof. Let f be a homomorphism from F to Γ such that the range f (F ) of f is maximal. If f is injective, we are done, otherwise F has elements u and v such that f (u) = f (v). Let A be the structure induced by f (F ) \ {f (u)} in Γ, and let B 1 and B 2 be two disjoint copies of the structure induced by f (F ) in Γ. Let e 1 be the embedding of A into B 1 that maps an element of f (F ) \ {f (u)} to its copy in B 1 . Similarly, there is an embedding e 2 : A → B 2 that maps an element of f (F ) \ {f (u)} to its copy in B 2 . By strong amalgamation of the age of Γ, there exist embeddings f 1 : B 1 → Γ and f 2 : Then the mapping f ′ : F → Γ defined by f ′ (w) = f 1 (e 1 (f (w))) if w = u, and defined by f ′ (w) = f 2 (e 2 (f (w))) if w = v, is well-defined. To see that it is a homomorphism, note that when R(x 1 , . . . , x n ) holds in F , then at most one of the x i can be mapped to f (u) since the tuples of R in Γ have only pairwise distinct entries. Since f (x) = f (y) implies that f ′ (x) = f ′ (y), and since moreover f ′ (u) = f ′ (v), the function f ′ also has a larger range than f , a contradiction.
The following is the central lemma connecting the Fraisse-limit of C 1 ∧ C 2 with the full product of the Fraisse-limits of C 1 and C 2 , so that we can ultimately use the product Ramsey theorem.
Lemma 3.4. Let C 1 and C 2 be strong amalgamation classes of ordered structures, with disjoint signatures σ and τ , such that all relations of C 1 and all relations of C 2 are injective. Let Γ be the Fraïssé-limit of C 1 ∧ C 2 with domain D, and suppose that Γ is ω-categorical. Let Γ 1 and Γ 2 be the σ-and τ -reduct of Γ, respectively. If Γ 1 and Γ 2 are cores, then the following structures are isomorphic.
Proof. It is straightforward to verify that d → (d, d) is an isomorphism between Γ and the substructure of To find an isomorphism between Γ and the model-complete core of Γ 1 ⊠ Γ 2 , it suffices to show that Γ is a model-complete core, and that Γ is homomorphically equivalent to Γ 1 ⊠ Γ 2 . We then use that the model-complete core is unique up to isomorphism (Theorem 2.1), which gives us the desired isomorphism. Model-completeness of Γ follows from homogeneity. To show that Γ is a core, let e be an endomorphism of Γ. Then e is an endomorphism of the σ-reduct Γ 1 of Γ, and an endomorphism of the τ -reduct of Γ 2 of Γ. Since both Γ 1 and Γ 2 are cores, e must be an embedding of Γ into Γ, which is what we wanted to show.
We finally show that Γ 1 ⊠Γ 2 and Γ are homomorphically equivalent. For one direction, recall that Γ maps to Γ 1 ⊠ Γ 2 via the mapping d → (d, d). For the other direction, it suffices to show that every finite substructure F of Γ 1 ⊠ Γ 2 homomorphically maps to Γ, by a standard compactness argument and ω-categoricity of Γ (see e.g. Lemma 3.1.5 in [2]). By Lemma 3.3, there is an injective homomorphism h 1 from the σ-reduct of F to Γ 1 (recall here that the order of Γ 1 is by assumption strict). Similarly, there is an injective homomorphism h 2 from the τ -reduct of F into Γ 2 . Let U be the (σ ∪ τ )structure with the same domain as F , and with relations defined as follows: for each R ∈ σ of arity k, a k-tuple t of elements of U is in R U if and only if h 1 (t) is in R Γ 1 . Similarly we define R U for relations R ∈ τ , with h 2 taking the role of h 1 and Γ 2 taking the role of Γ 1 . Clearly, h 1 is an embedding of the σ-reduct U 1 of U into Γ 1 , and h 2 is an embedding of the τ -reduct U 2 of U into Γ 2 . Therefore, U 1 ∈ C 1 and U 2 ∈ C 2 . By definition of C := C 1 ∧ C 2 , we have that U ∈ C, and there is an embedding e of U into Γ. Then e is the desired homomorphism from F to Γ.
Proof of Theorem 1.5. Let Γ be the Fraïssé-limit of C 1 ∧ C 2 , and let Γ 1 and Γ 2 be the σ-and τ -reduct of Γ, respectively. Note that Γ 1 and Γ 2 are also homogeneous, and ω-categorical by assumption. Moreover, since C 1 and C 2 have the Ramsey property, by Theorem 1.9 there are linear orders < Γ 1 and < Γ 2 with quantifier-free first-order definitions ϕ 1 and ϕ 2 in Γ 1 and Γ 2 , respectively.
Let Γ * 1 be the structure with the same domain as Γ 1 whose relations are exactly the injective relations that are first-order definable in Γ 1 . Note that this includes in particular the linear order < Γ 1 , and so Γ * 1 is ordered. Also note that Γ * 1 is homogeneous and a core, and has the same (oligomorphic) automorphism group as Γ 1 , since Γ * 1 contains an n-ary relation for each orbit of n-tuples of distinct elements from Γ. Moreover, observe that algebraic closure only depends on the automorphism group, and it follows by Proposition 1.1 that that the age of Γ * 1 has strong amalgamation. We write σ * for the signature of Γ * 1 . Analogously we define the structure Γ * 2 from Γ 2 ; we choose the signature τ * for Γ * 2 such that τ * is disjoint from σ * . Finally, let Γ * be the (τ * ∪ σ * )-structure obtained as the common expansion of Γ * 1 and Γ * 2 . Then Aut(Γ * ) = Aut(Γ), because an orbit of an ntuple t in Γ is uniquely given by the orbit of t in Γ * 1 and the orbit of t in Γ * 2 . In particular, the automorphism group of Γ * is oligomorphic, and by the theorem of Ryll-Nardzewski Γ * is ω-categorical. Moreover, Γ * is homogeneous and therefore the Fraïssé-limit of its age. The groups Aut(Γ * 1 ) = Aut(Γ 1 ) and Aut(Γ * 2 ) = Aut(Γ 2 ) are oligomorphic, and extremely amenable by Theorem 1.9. By Proposition 3.1, Γ * 1 ⊠ Γ * 2 is homogeneous, and by Proposition 3.2, Aut(Γ * 1 ⊠ Γ * 2 ) is extremely amenable. Then Theorem 2.3 (again in combination with Theorem 1.9) shows that the model-complete core of Γ * 1 ⊠ Γ * 2 has an extremely amenable automorphism group G. By Lemma 3.4, the model-complete core of Γ * 1 ⊠ Γ * 2 is isomorphic to Γ * , and hence Aut(Γ * ) = Aut(Γ) is extremely amenable. We conclude by Theorem 1.9 that C 1 ∧ C 2 is Ramsey.

Forgetting one order
We finally prove Proposition 1.6: let C 1 and LO ∧ C 2 be Ramsey classes with strong amalgamation and ω-categorical Fraïssé-limits, and suppose that C 1 , C 2 , and LO have pairwise disjoint relational signatures. We have to show that C 1 ∧ C 2 has the Ramsey property.
Proof of Proposition 1.6. We use the fact that C 3 := C 1 ∧ (LO ∧ C 2 ) is Ramsey by Theorem 1.5. Let Γ be the Fraïssé-limit of C 1 . By Theorem 1.9, there is a linear order < on the elements of Γ that has a quantifier-free first-order definition ϕ(x, y) in Γ.
To show that C 1 ∧ C 2 is Ramsey, let A and B be from C 1 ∧ C 2 . Let A ′ , B ′ be the expansion of A, B by the relation < defined by ϕ over A and B, respectively. Note that A ′ , B ′ ∈ C 3 . Since C 3 is Ramsey, there exists a C ′ ∈ C 3 such that C ′ → (B ′ ) A ′ r . Let C be the reduct of C ′ where we drop the relation <. We claim that C → (B) A r . Let χ : C A → [r] be arbitrary. We define a coloring χ ′ : C ′ A ′ → [r] as follows. Let e be an arbitrary embedding of A ′ into C ′ . Since A is a reduct of A ′ , and C is a reduct of C ′ with the same signature, the mapping e is also an embedding of A into C. Therefore, e is in the range of χ, and we can define χ ′ (e) := χ(e). Since C ′ → (B ′ ) A ′ r , there exists an f ∈ C ′ B ′ such that χ ′ is constant c on f [B ′ ] A ′ . By the same argument as above, f is also an embedding of B into C. We claim that χ is constant on f [B] A . Let e be an arbitrary embedding of A into f [B]. Recall that A ′ and B ′ are the expansion of A and B by the relation < defined by ϕ. Since embeddings preserve quantifier-free formulas, e preserves in particular ϕ. Therefore, the mapping e is an embedding of A ′ into the substructure f [B ′ ] of C ′ . In particular, e is in the range of χ ′ , and χ ′ (e) = c. It follows that χ(e) = c, which concludes the proof that χ is constant on f [B] A .