aa r X i v : . [ m a t h . L O ] A p r CORES OVER RAMSEY STRUCTURES
ANTOINE MOTTET AND MICHAEL PINSKER
Abstract.
It has been conjectured that the class of first-order reducts of finitely boundedhomogeneous Ramsey structures enjoys a CSP dichotomy; that is, the Constraint SatisfactionProblem of any member of the class is either NP-complete or polynomial-time solvable. Thealgebraic methods currently available that might be used for confirming this conjecture,however, only apply to structures of the class which are, in addition, model-complete cores.We show that the model-complete core associated with any member of this class again belongsto the class, thereby removing that obstacle.Our main result moreover answers several open questions about Ramsey expansions: inparticular, if a structure has an ω -categorical Ramsey expansion, then so do its model com-panion and its model-complete core. Introduction
Constraint Satisfaction Problems and two dichotomy conjectures.
The
Con-straint Satisfaction Problem (CSP) of a relational structure A in a finite signature, denotedby CSP( A ), is the computational problem of deciding its primitive positive theory; that is,given a primitive positive sentence in the language of A , one has to decide whether or not thesentence holds in A .For the class of finite structures, constraint satisfaction problems turn out to enjoy a complexity dichotomy : for every finite structure A , the problem CSP( A ) is either NP-completeor polynomial-time solvable. This was shown in 2017 independently by Bulatov [20] andZhuk [33], confirming a conjecture of Feder and Vardi [23] that had remained unresolvedfor 25 years.A similar conjecture was formulated in 2011 by Bodirsky and the second author (see [18]) fora significant expansion of the class of finite structures: namely, the class of first-order reductsof finitely bounded homogeneous structures . This wider conjecture has since been confirmedfor numerous subclasses [9, 15, 29, 8, 11, 12, 10]. Here, we call a structure homogeneous ifevery isomorphism between finite induced substructures extends to an automorphism of theentire structure; it is finitely bounded if it has a finite signature and the set of its finite inducedsubstructures is given by a finite number of forbidden substructures. A first-order reduct isa structure first-order definable without parameters.1.2. Model-complete cores.
Different structures A , B in the same signature can have thesame CSP, i.e., they can satisfy the same primitive positive sentences. For finite structures,this is the case if and only if they are homomorphically equivalent , i.e., there exists a homo-morphism from A to B and vice-versa. The same statement holds for ω -categorical structures, Antoine Mottet has received funding from the European Research Council (ERC) under the EuropeanUnions Horizon 2020 research and innovation programme (grant agreement No 771005). Michael Pinsker hasreceived funding from the Austrian Science Fund (FWF) through project No P32337, and from the CzechScience Foundation (grant No 13-01832S). i.e., countable structures whose automorphism group acts with only finitely many orbits ontuples of any fixed finite length. The class of ω -categorical structures largely contains that offirst-order reducts of finitely bounded homogeneous structures.In the context of CSPs, it is thus natural to consider structures up to homomorphic equiva-lence, and to raise the question of the existence of a “nicest” representative of each equivalenceclass. Note that the range of any endomorphism of a structure A induces a homomorphicallyequivalent structure in A . For a finite structure A , iterating this argument yields a “smallest”representative A ′ of its equivalence class whose every endomorphism is an automorphism.Structures with the latter property are called cores [25], and the representative A ′ is calledthe core of A ; it is uniquely defined, up to isomorphism, by the homomorphic equivalence to A and its coreness.For ω -categorical structures, finding the smallest representative is much less obvious. Oneof the first achievements in the theory of CSPs of such structures was the discovery of theappropriate notion [6]: a structure is called a model-complete core if its automorphisms aredense in its endomorphisms in the topology of pointwise convergence; that is, every endo-morphism agrees, on every finite subset of its domain, with some automorphism. Both theoriginal proof from [6] and the later proof in [1, 2] of the following theorem require a delicatecombination of Fra¨ıss´e-type and compactness arguments. Theorem 1 (Bodirsky [6]) . Let A be an ω -categorical structure. Then A is homomorphicallyequivalent to a model-complete core A ′ . Moreover, A ′ is again ω -categorical and unique up toisomorphism. The algebraic approach to CSPs assigns to every relational structure A an algebra on itsdomain whose functions are the polymorphisms of A , that is, the homomorphisms from finitepowers of A to A . For finite A , it is known that the complexity of CSP( A ) only depends onthe identities of its polymorphisms, that is, the universally quantified equations that holdbetween them. The dichotomy proofs of Bulatov and Zhuk then show that, as had beenconjectured in [21], either there is a non-trivial identity satisfied by the polymorphisms of thecore of a finite structure, and its CSP is polynomial-time solvable, or otherwise its CSP is NP-complete. The algebraic approach can be adapted for ω -categorical structures [16]. Resultsfrom [4, 5] then lead to the following formulation of the conjecture of Bodirsky and the secondauthor, which proposes a precise algebraic borderline between hardness and tractability forCSPs of first-order reducts of finitely bounded homogeneous structures. A pseudo-Siggers polymorphism is a polymorphism satisfying a certain identity. Conjecture 2.
Let A be a first-order reduct of a finitely bounded homogeneous structure, andlet A ′ be its model-complete core. Then: • either A ′ has no pseudo-Siggers polymorphism, and CSP( A ) is NP-complete, or • A ′ has a pseudo-Siggers polymorphism, and CSP( A ) is polynomial-time solvable. It is known that if an ω -categorical structure has no pseudo-Siggers polymorphism, thenits CSP is NP-hard [4, 2]. Therefore, to prove Conjecture 2, we need to show that if themodel-complete core A ′ of a first-order reduct A of a finitely bounded homogeneous structurecontains has a pseudo-Siggers polymorphism, then its CSP is polynomial-time solvable. Thefollowing observations are crucial in this context. • We cannot weaken the assumption on A to ω -categoricity: there are examples of ω -categorical structures with even undecidable CSPs whose model-complete core has a ORES OVER RAMSEY STRUCTURES 3 pseudo-Siggers polymorphism. We refer to [24] for a hierarchy of hard CSPs withnumerous desirable algebraic and model-theoretic properties. • We cannot consider the pseudo-Siggers criterion in the structure A instead of itsmodel-complete core A ′ : there exist even finite structures with a pseudo-Siggers poly-morphism and an NP-complete CSP [4].It follows that with Conjecture 2, we face the dilemma of having to combine a model-theoreticproperty of one structure ( A ) with an algebraic property of another ( A ′ ). While we knowthat A ′ inherits ω -categoricity from A , which is insufficient to prove tractability of the CSP,we do not know if this is the case for the stronger assumptions on A . This raises the followingquestion, a positive answer to which would allow us to assume A = A ′ . We believe it was firstasked by Bodirsky at the Ordener Lectures, Paris, in 2012. Question 3.
Is the class of first-order reducts of a finitely bounded homogeneous structuresclosed under taking model-complete cores?
We remark that there exists an alternative formulation of Conjecture 2 which successfullyavoids model-complete cores via height 1 identities of polymorphisms [2]. However, thatformulation does not provide any concrete identity (similar to the one defining a pseudo-Siggers polymorphism) which could be used for proving the conjecture, making the proposedtractability criterion difficult to apply; in fact, it has recently been shown that no fixed setof height 1 identities can characterize polynomial-time solvability for structures within thescope of the conjecture [13]. This is in contrast with the finite case, where polynomial-timesolvability can be characterized by the presence of a
Siggers polymorphism (without the useof cores), by results from [3].1.3.
The Ramsey property.
All successful complexity classifications for subclasses of therange of Conjecture 2 use a method devised in [19, 14] which involves Ramsey theory to reducethe problem to finite structures. In particular, the proofs draw on the fact that the structuresunder consideration are first-order reducts of a finitely bounded homogeneous structure whichhas a finitely bounded homogeneous
Ramsey expansion : it has an expansion by some relationswhich is still finitely bounded, homogeneous, and
Ramsey , i.e., enjoys a certain combinatorialproperty which, roughly, ensures the existence of monochromatic substructures in specificcolorings. In fact, the second author of the present article, also an author of the conjecture,claims that the conjecture was formulated without mentioning the Ramsey property onlybecause it was believed that such a Ramsey expansion always exists.
Question 4.
Does every finitely bounded homogeneous structure have a finitely bounded homo-geneous Ramsey expansion? In other words, is every first-order reduct of a finitely boundedhomogeneous structure also a first-order reduct of a finitely bounded homogeneous Ramseystructure?
This question has, in various formulations and with varying scope, been considered byseveral authors: by Bodirsky and the second author when formulating the dichotomy conjec-ture, and also in the context of a decidability result for first-order reducts of finitely boundedhomogeneous Ramsey structures [19]; and later by Melleray, Van Th´e, and Tsankov [30] inthe context and language of topological dynamics. In [7], Bodirsky formulates a conjectureclaiming a positive answer for homogeneous structures in a finite language, and extensivelyargues the importance of this conjecture. Neˇsetˇril addressed the question indirectly in thecontext of the characterisation of Ramsey classes [31]. Hubiˇcka and Neˇsetˇril then obtained
ANTOINE MOTTET AND MICHAEL PINSKER a positive answer for an impressive number of structures [27, 26]. In [22], Evans, Hubiˇckaand Neˇsetˇril gave an example of an ω -categorical structure without ω -categorical Ramseyexpansion, but the problem as formulated in Question 4 remains open.Given the importance of Ramsey expansions, it is natural to investigate how the existenceof a Ramsey expansion is preserved under various constructions. This question was the maintheme of the survey [7], where one of the results is that if A is itself a homogeneous Ramseystructure, then its model-complete core is homogeneous and Ramsey. The same is true for model-companions . An ω -categorical structure A is model-complete if its automorphisms aredense in its self-embeddings; equivalently, this is the case if and only if every self-embeddingof A is an elementary map. We say that A ′ is a model-companion of A if it is model-completeand A embeds into A ′ and vice-versa. The existence of an ω -categorical model-companion forevery ω -categorical structure is a classical result of Saracino [32] from 1973 but it is subsumedby Theorem 1, as we will see in Section 2 below. The following problems were left open in [7]. Question 5 (Questions 7.1, 7.2 in [7]) . Let A be a structure. (1) Suppose that A has a homogeneous Ramsey expansion with finite signature. Does themodel-complete core of A (resp., its model-companion) have such an expansion? (2) Suppose that A has an ω -categorical Ramsey expansion. Does the model-complete coreof A (resp., its model-companion) have such an expansion? Results
Our main theorem is the following.
Theorem 6.
Let A be a first-order reduct of an ω -categorical homogeneous Ramsey structure B , and let A ′ be its model-complete core. Then: • A ′ also is a first-order reduct of an ω -categorical homogeneous Ramsey structure B ′ that is a substructure of B . • If B is finitely bounded, then B ′ can be chosen to be finitely bounded as well. This provides a positive answer to Question 5 for model-complete cores. As a by-product,we obtain the same answer for the variant with model-companions. Indeed, given any ω -categorical structure A , let C be the expansion of A by the complement of each of its relations.Let C ′ be the model-complete core of C , and let A ′ be the reduct of C ′ obtained by forgettingthe new relations. Then A ′ is model-complete since it has the same embeddings as C ′ . Anyhomomorphism from C to C ′ has to be an embedding since the complement of every relationhas to be preserved, so that there exists an embedding of A into A ′ , i.e, A ′ is the model-companion of A . Thus, if A has a homogeneous Ramsey expansion (with finite signature),so does C , which means by Theorem 6 that C ′ has a homogeneous Ramsey expansion, andfinally this implies that A ′ has a homogeneous Ramsey expansion.We remark that the results of [7] mentioned before Question 5 (Theorems 3.15 and 3.18in [7]) are a corollary of the proof of Theorem 6, in the special case that A = B .Since Theorem 6 is also compatible with the additional condition of finite boundedness,this implies that we obtain a positive answer to Question 3 provided Question 4 has a positiveanswer: that is, we obtain that the class of first-order reducts of finitely bounded homogeneousRamsey structures is closed under taking model-complete cores.It might be interesting to note that our proof does not use Theorem 1. Rather than that,it refines, and perhaps sheds some light on, the proof of Theorem 1 in [2]. ORES OVER RAMSEY STRUCTURES 5 Preliminaries
We use blackboard boldface letters such as A , B , . . . for relational structures, and the sameletters in plain font A, B, . . . for their domains. Similarly, we write M , N , . . . for transforma-tion semigroups, in particular for permutation groups, and M, N, . . . for their domains. Alldomains of relational structures as well as of transformation semigroups are tacitly assumedto be countable.Let G be a permutation group. Then G naturally acts componentwise on G n , for all n ≥ orbit of G ; wewill sometimes use the notion n -orbit when we wish to specify the action. A permutationgroup G is oligomorphic if it has finitely many n -orbits for every n ≥ A ) of any relational structure A is a permutation group, and A is ω -categorical if Aut( A ) is oligomorphic. The group Aut( A ) is a closed subgroup of the fullsymmetric group on its domain A with respect to the topology of pointwise convergence , i.e.,the product topology on A A where A is taken to be discrete. It is called extremely amenable if any continuous action on a compact Hausdorff space has a fixed point.The endomorphism monoid End( A ) of a relational structure A is a transformation monoidon its domain A . This monoid also acts naturally componentwise on finite powers of A , andwe shall write t ( a ) for the n -tuple obtained by applying t ∈ End( A ) to a tuple a ∈ A n , forany n ≥
1. The monoid End( A ) also bears the topology of pointwise convergence, and A iscalled a model-complete core if Aut( A ) is dense in End( A ) with respect to this topology. If M is a subset of M M (e.g., a transformation semigroup, or even a permutation group on M ),then we write M for the closure of M in M M (and not in the symmetric group on M , evenif M is a permutation group!) with respect to the topology of pointwise convergence. Thus,a function f : M → M is an element of M if for every finite F ⊆ M there exists g ∈ M suchthat g agrees with f on F . A transformation semigroup M is closed if M = M .If M , N are transformation semigroups on the same domain, then M is left-invariant(right-invariant) under N if n ◦ m ∈ M ( m ◦ n ∈ M ) for all n ∈ N and all m ∈ M . Thesemigroup M is invariant under N if it is left- and right-invariant. A left-ideal of M is asubsemigroup which is left-invariant under M .Two structures A , A ′ are homomorphically equivalent if there exists a homomorphism from A to A ′ and vice-versa. Whenever A is a relational structure, and A ′ , A ′′ are model-completecores which are homomorphically equivalent to A , then A ′ , A ′′ are isomorphic. We can thusspeak of the model-complete core of a structure A if it exists.A structure A is homogeneous if every isomorphism between finite induced substructuresof A extends to an automorphism of A . If A is homogeneous, then its age , i.e., the class ofits finite induced substructures up to isomorphism, has the amalgamation property (AP) : aclass K of structures has the AP if for all A , A , A ∈ K and all embeddings e , e of A into A , A , respectively, there exist embeddings f , f of A , A into a structure C ∈ K such that f ◦ e = f ◦ e . Conversely, if the age K of some countable structure A ′ has the AP, thenthere exists a countable homogeneous structure A whose age is equal to K . The structure A is called the Fra¨ıss´e limit of K .A class K of finite structures in a common finite signature is called finitely bounded if thereexists a finite set F of structures in that signature such that membership in K is equivalentto not embedding any member of F . An infinite structure is called finitely bounded if its ageis finitely bounded. ANTOINE MOTTET AND MICHAEL PINSKER
Given two structures S , F , an isomorphic copy of S in F is an embedding from S to F . Astructure A is Ramsey if for all structures S , F in its age and all colorings of the isomorphiccopies of S in A with two colors there exists an isomorphic copy of F in A on which thecoloring is constant. If A is homogeneous, then this is the case if and only if Aut( A ) isextremely amenable [28]. 4. The Proof
Range-rigidity.Definition 7.
Let G be a permutation group, and let g : G → G be a function. We call g range-rigid with respect to G if for all β ∈ G we have g ∈ { α ◦ g ◦ β ◦ g | α ∈ G } ; in otherwords, every orbit of G which has a tuple within the range of g is invariant under g . Lemma 8.
Let B be a homogeneous structure, and let g : B → B be range-rigid with respectto Aut( B ) . Then the age of the structure induced by the range of g in B has the AP.Proof. Let
U, V, W ⊆ B be so that g [ U ] , g [ V ] , g [ W ] are finite, and such that the structure S U induced by g [ U ] embeds into the structures S V and S W induced by g [ V ] and g [ W ], respectively.By the homogeneity of B , we know that these embeddings can be performed by restrictingautomorphisms α, β ∈ Aut( B ) to g [ U ]. Consider the structure F induced by F := g [ α − [ g [ V ]] ∪ β − [ g [ W ]] in B . Then F is an amalgam of S V and S W over S U and embeddings given by α, β :the witnessing embeddings of the amalgamation are the restrictions of g ◦ α − and g ◦ β − to g [ V ] and g [ W ], respectively: the fact that these restrictions are embeddings follows from therange-rigidity of g . (cid:3) Definition 9.
Let B be an ω -categorical homogeneous structure, and let g : B → B be range-rigid with respect to Aut( B ) . • We denote by B g the Fra¨ıss´e limit of the class of finite structures induced by the rangeof g (which has the AP, by Lemma 8). By the homogeneity of B , we may assume that B g is an induced substructure of B . • If A is a first-order reduct of B , then we denote by A g the substructure induced by thedomain of B g in A . We remark that in Definition 9, since B is homogeneous and ω -categorical, A has aquantifier-free first-order definition in B . Hence, A g is well-defined, i.e., independent of theembedding of B g into B .In the following three lemmas, we show that B g retains the properties of B that we areinterested in. Lemma 10.
Let B be a homogeneous structure, and let g : B → B be range-rigid with respectto Aut( B ) . If B is ω -categorical, then so is B g .Proof. Since B g is homogeneous, it suffices to prove that for every n ≥
1, there are only finitelymany atomic formulas with n free variables modulo equivalence. Since B g is a substructureof B , if two atomic formulas are equivalent in B then they are equivalent in B g . Since B ishomogeneous and ω -categorical, for every n ≥ n free variables in B , and we get the desired result. (cid:3) Lemma 11.
Let B be a homogeneous structure, and let g : B → B be range-rigid with respectto Aut( B ) . If B is finitely bounded, then so is B g . ORES OVER RAMSEY STRUCTURES 7
Proof.
Let F be a finite set of forbidden substructures for the age of B . Let m ≥ B . Let F ′ consist of all structures in F , plus allstructures on the set { , . . . , m } which are isomorphic to a substructure of B but not of B g .Clearly, if a finite structure in the signature of B embeds a member of of F ′ , then it cannotbe in the age of B g . Conversely, if such a structure F does not embed any member of F ′ ,then it embeds into B , hence we may assume it is a substructure of B . Applying g to thissubstructure, we obtain a structure isomorphic to F , because g preserves all m -orbits whichintersect its range, and all m -element substructures of F belong to such orbits. Hence, g shows that F embeds into B g . (cid:3) Lemma 12.
Let B be a homogeneous structure, and let g : B → B be range-rigid with respectto Aut( B ) . If B is Ramsey, then so is B g .Proof. Let S , F be finite induced substructures of B g , and let χ be a coloring of the isomorphiccopies of S in B g with two colors. Let f : g [ B ] → B g be a function which is an embeddingwith respect to the relations of B ; such an embedding exists by the homogeneity of B g andsince the age of the structure induced by g [ B ] in B is equal to the age of B g . The coloring χ then induces a coloring χ ′ of the isomorphic copies of S in B , by precomposing χ with f ◦ g .Since B is Ramsey, there exists an isomorphic copy of F in B on which χ ′ is constant. Theimage of F under f ◦ g then is a substructure of B g on which the coloring χ is constant; it isisomorphic to F since g is range-rigid. (cid:3) Lemma 13.
Let B be an ω -categorical homogeneous structure, and let A be a first-orderreduct of B . Suppose that N ⊆ End( A ) is a minimal closed left ideal, and that g ∈ N isrange-rigid with respect to Aut( B ) . Then A g is the model-complete core of A .Proof. We first prove that the set I := { ( a, b ) | ∃ n ≥ ∃ e ∈ End( A g ) ( a, b ∈ ( A g ) n ∧ e ( a ) = b ) } is a back-and-forth system of partial isomorphisms of A g . This implies that A g is a model-complete core, since we then obtain for every e ∈ End( A g ) and every finite tuple a an auto-morphism α ∈ Aut( A g ) such that α ( a ) = b . To prove that I consists of partial isomorphismsof A g and has the back-and-forth property, note that it suffices to show the following: forevery e ∈ End( A g ) and every finite tuple a of elements in A g , there exists e ′ ∈ End( A g ) suchthat e ′ ( e ( a )) = a .To this end, let f : g [ B ] → B g be a function which is an embedding with respect to therelations of B , and such that a lies within the range of f ◦ g . The existence of f is ensuredby the homogeneity of B g and by the fact that the ages of the structures induced by g [ B ]and by g [ B ] in B both equal the age of B g . We then have that e ◦ f ◦ g ∈ End( A ). Let b be a tuple such that a = ( f ◦ g )( b ). By the fact that g lies in a minimal closed left ideal ofEnd( A ), there exists h ∈ End( A ) such that h ◦ ( e ◦ f ◦ g )( g ( b )) = g ( b ) . Hence, a = ( f ◦ g )( b ) = ( f ◦ g )( g ( b )) = ( f ◦ g ) ◦ h ◦ ( e ◦ f ◦ g )( g ( b )) = ( f ◦ g ◦ h ) ◦ e ( a ) . The restriction of the function f ◦ g ◦ h ∈ End( A ) to A g thus bears witness to the statementwe wanted to prove. ANTOINE MOTTET AND MICHAEL PINSKER
Since A g is a substructure of A , and since f ◦ g is a homomorphism from A to A g , thestructures A g and A are homomorphically equivalent. Whence, A g is indeed the model-complete core of A . (cid:3) From extreme amenability to canonicity to range-rigidity.
The following defi-nition from [19] is incomparable to range-rigidity, but will allow us to produce range-rigidfunctions.
Definition 14.
Let G be a permutation group, and let g : G → G be a function. We call g canonical with respect to G if for all β ∈ G we have g ∈ { α ◦ g ◦ β | α ∈ G } ; in other words,the image of any orbit of G under g is contained in an orbit. Lemma 15 (The canonisation lemma [19]; cf. [17]) . Let G be a closed oligomorphic extremelyamenable permutation group, and let g : B → B . Then { α ◦ g ◦ β | α, β ∈ G } contains a canonical function with respect to G . Lemma 16.
Let G be a closed oligomorphic permutation group, and let M ⊆ G G be anon-empty closed transformation semigroup which is invariant under G and which containsa canonical function with respect to G . Then M contains a range-rigid function with respectto G .Proof. Pick any canonical function g ∈ M . For every n ≥
1, there exist k n ≥ g k n ◦ g k n [ O ] ⊆ g k n [ O ] for all n -orbits O of G . Let ∼ be the equivalence relation on G G defined by f ∼ f ′ if f ′ ∈ { α ◦ f | α ∈ G } . It is known that G G / ∼ is compact (see, e.g.,Lemma 4 in [17]). The sequence ([ g k n ] ∼ ) n ≥ thus has an accumulation point in G G / ∼ , whichmeans that there exists a sequence ( α n ) n ≥ of elements of G such that ( α n ◦ g k n ) n ≥ has anaccumulation point h in G G . We then have that h [ O ] ⊆ h [ O ] for all orbits O of G . Since h is also canonical with respect to G , this means that h ∈ M is range-rigid with respect to G . (cid:3) Lemma 17.
Let M be a closed transformation monoid containing a closed oligomorphicextremely amenable permutation group G . Then M contains a minimal closed left idealwhich contains a range-rigid function with respect to G .Proof. The fact that M contains a minimal closed left ideal N can be proved by a standardcompactness argument (see [1, 2] for the proof).Pick any g ∈ N , and let S be the smallest non-empty closed transformation semigroupwhich contains g and which is invariant under G . By Lemma 15, S contains a canonicalfunction with respect to G , and hence by Lemma 16, S contains a range-rigid function withrespect to G . It is easy to see that any element of S belongs to a minimal closed left idealof M . (cid:3) Summary of the proof of Theorem 6.
Proof of Theorem 6.
Applying Lemma 17 to the monoid M := End( A ) and the group G :=Aut( B ), we get that End( A ) contains a minimal closed left ideal which contains a range-rigidfunction g with respect to Aut( B ). By Lemma 13, A g is the model-complete core of A . Bythe ω -categoricity and the homogeneity of B , the first-order reduct A has a quantifier-freedefinition in B . Hence, A g has a quantifier-free definition in B g , and in particular is a first-order reduct thereof. Lemma 10 gives that B g is ω -categorical, while Lemma 12 tells us that ORES OVER RAMSEY STRUCTURES 9 B g is a homogeneous Ramsey structure. Finally, by Lemma 11 it is finitely bounded if B is. (cid:3) References [1] Libor Barto, Michael Kompatscher, Miroslav Olˇs´ak, Trung Van Pham, and Michael Pinsker. The equiva-lence of two dichotomy conjectures for infinite domain constraint satisfaction problems. In
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Department of Algebra, MFF UK, Sokolovsk´a 83, 186 00 Praha 8, Czech Republic
E-mail address : [email protected] URL : Institut f¨ur Diskrete Mathematik und Geometrie, FG Algebra, TU Wien, Austria, and De-partment of Algebra, Charles University, Czech Republic
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