Ramsey properties and extending partial automorphisms for classes of finite structures
RRamsey properties and extending partialautomorphisms for classes of finite structures
David M. Evans ∗ , Jan Hubiˇcka † , Jaroslav Neˇsetˇril † Abstract
We show that every free amalgamation class of finite structures withrelations and (symmetric) partial functions is a Ramsey class when en-riched by a free linear ordering of vertices. This is a common strength-ening of the Neˇsetˇril-R¨odl Theorem and the second and third authors’Ramsey theorem for finite models (that is, structures with both relationsand functions). We also find subclasses with the ordering property. Forlanguages with relational symbols and unary functions we also show theextension property for partial automorphisms (EPPA) of free amalgama-tion classes. These general results solve several conjectures and providean easy Ramseyness test for many classes of structures.
In this paper we discuss three related concepts — Ramsey classes, the ordering(or lift or expansion) property and the extension property for partial auto-morphisms (EPPA). The main novelty of our results is that they hold for freeamalgamation classes of finite structures with both relations and partial func-tions . This provides a useful tool for proving these types of results for someclasses of structures which naturally carry a closure operation. We will explainbelow what we mean by this; examples and applications are given at the end ofthe paper.As is well known, all three of these concepts about classes of finite structuresare related to issues in topological dynamics and this relationship provides muchof the motivation for what we do. For example, by [22] the automorphism groupof the Fra¨ıss´e limit of a Ramsey class R is extremely amenable. Moreover, if theRamsey class R has the ordering property with respect to some amalgamationclass K , then it determines the universal minimal flow of the automorphism ∗ Department of Mathematics, Imperial College London, London SW7 2AZ, UK, [email protected] † Supported by grant ERC-CZ LL-1201 of the Czech Ministry of Education and CE-ITI P202/12/G061 of GAˇCR. Computer Science Institute of Charles University (IUUK),Charles University, Malostransk´e n´am. 25, 11800 Praha, Czech Republic, 118 00 Praha 1, { hubicka , nesetril } @iuuk.mff.cuni.cz a r X i v : . [ m a t h . C O ] D ec roup of the Fra¨ıss´e limit of K . Thus, our results Theorem 1.3 and Theorem 1.4about the Ramsey and ordering properties give new examples of this correspon-dence. By [23] and our Theorem 1.7 about EPPA, the automorphism groupof the Fra¨ıss´e limit of every free amalgamation class K in a language where allfunctions are unary is amenable. However we note that the same conclusion,without the restriction that the partial functions are unary, also follows fromour Ramsey Theorem 1.3 and Proposition 9.3 of [2] (note that the assumptionof OP in the statement of Proposition 9.3 is not necessary, as was observed byPawliuk and Soki´c in [34]).To generalise naturally the known results about relational structures, weneed to define carefully what we mean by a structure and substructure, be-cause none of these results holds in the context of free amalgamation classeswith strong embeddings (as discussed in [11], see also the remarks precedingTheorem 1.3). In the Ramsey theory setting it is common to work with ‘in-complete’ structures. Thus we have to modify the standard model-theoreticnotion of structures (see e.g. [15]), where functions are required to be total andthus complete in some sense. Before stating the main results, we give the basic(model-theoretic) setting of this paper. We introduce a variant of the usualmodel-theoretic structures which allows partial functions and which is well tai-lored to the Ramsey setting.Let L = L R ∪ L F be a language with relational symbols R ∈ L R and functionsymbols F ∈ L F each having associated arities denoted by a ( R ) for relationsand domain arity , d ( F ), range arity , r ( F ), for functions. Denote by (cid:0) An (cid:1) the setof all subsets of A consisting of n elements. An L -structure A is a structurewith vertex set A , functions F A : Dom( F A ) → (cid:0) Ar ( F ) (cid:1) , Dom( F A ) ⊆ A d ( F ) , F ∈ L F and relations R A ⊆ A a ( R ) , R ∈ L R . The set Dom( F A ) is called the domain of the function F in A . Notice that the domain is a set of ordered d ( F )-tuples while the range is set of unordered r ( F )-tuples. Symmetry in theranges permits an explicit description of algebraic closures in the Fra¨ıss´e limitswithout changing the automorphism group. It also greatly simplifies some ofthe notation below.The language is usually fixed and understood from the context (and it is inmost cases denoted by L ). If the set A is finite we call A a finite structure . Weconsider only structures with finitely or countably infinitely many vertices. Ifthe language L contains no function symbols, we call L a relational language and say that an L -structure is a relational L -structure . A function symbol F such that d ( F ) = 1 is a unary function .The notions of embedding, isomorphism, homomorphisms and free amal-gamation classes are natural generalisations of the corresponding notions onrelational structures and are formally introduced in Section 1.4. Consideringfunction symbols has important consequences for what we consider a substruc-ture. An L -structure A is a substructure of B if A ⊆ B and all relations andfunctions of B restricted to A are precisely relations and functions of A . In2articular if some n -tuple (cid:126)t of vertices of A is in Dom( F B ) then it is also inDom( F A ) and F A ( (cid:126)t ) = F B ( (cid:126)t ). This implies that B does not induce a substruc-ture on every subset of B (but only on ‘closed’ sets, to be defined later).Building on these standard model theoretic notions we now outline the con-tents of this paper. We proceed to the three main directions—Ramsey theory,the ordering property and the extension property for partial automorphisms(EPPA). In each of these directions we now state the main result. This can besummarised by saying that for free amalgamation classes we have strong pos-itive theorems in each of these areas. There is more to it than just meets theeye: for the first time we demonstrate affinity of all these directions (for theordering property and Ramsey this is known, but for EPPA much less so).This has a number of applications to special classes of structures. In Section 5we give several examples which have received recent attention: k -orientations,bowtie-free graphs and Steiner systems. These all are easy consequences of ourmain result. Finally, let us remark that our results further narrow the gap forthe project of characterisation of Ramsey classes [25]. For structures A , B denote by (cid:0) BA (cid:1) the set of all sub-structures of B , which areisomorphic to A . Using this notation the definition of a Ramsey class has thefollowing form. A class C is a Ramsey class if for every two objects A and B in C and for every positive integer k there exists a structure C in C such that thefollowing holds: For every partition (cid:0) CA (cid:1) into k classes there exists a (cid:101) B ∈ (cid:0) CB (cid:1) such that (cid:0) (cid:101) BA (cid:1) belongs to one class of the partition. It is usual to shorten thelast part of the definition to C −→ ( B ) A k .We are motivated by the following, now classical, result. Theorem 1.1 (Neˇsetˇril-R¨odl Theorem [27, 32]) . Let L be a relational language, ( A , ≤ ) and ( B , ≤ ) be ordered L -structures. Then there exists an ordered L -structure ( C , ≤ ) such that ( C , ≤ ) −→ ( B , ≤ ) ( A , ≤ )2 .Moreover, A and B do not contain irreducible ((see Section 2) L -structure F as an substructure then C may be chosen with the same property. In our setting this result may be reformulated as follows: Given a language L , denote by −→ L the language L extended by one binary relation ≤ . Given an L -structure A , an ordering of A is an −→ L -structure extending A by an arbitrarylinear ordering ≤ A of the vertices. For brevity we denote such ordered A as −→ A . Given a class K of L -structures, denote by −→K the class of all orderings ofstructures in K . We sometimes say that −→K arises by taking free orderings ofstructures in K . Note that minimal relational structures which do not belongto an free amalgamation class are all irreducible. Thus Theorem 1.1 can now bere-formulated using basic notions of Fra¨ıss´e theory (which will be introduced inSection 1.4) as follows: 3 heorem 1.2 (Neˇsetˇril-R¨odl Theorem for free amalgamation classes) . Let L be a relational language and K be a free amalgamation class of relational L -structures. Then −→K is a Ramsey class. The more recent connection between Ramsey classes and extremely amenablegroups [22] has motivated a systematic search for new examples. It becameapparent that it is important to consider structures with both relations andfunctions or, equivalently, classes of structures with “strong embeddings”. Thisled to [20] which provides a sufficient structural condition for a subclass of aRamsey class of structures to be Ramsey and also generalises this approachto classes with formally-described closures. Comparing the two main resultsof [20] (Theorem 2.1 for classes without closures and Theorem 2.2 for classeswith closures) it is clear that considering classes with closures leads to manytechnical difficulties. In fact, a recent example given by first author based onHrushovski’s predimension construction [11] not only answers one of the mainquestions in the area (about the existence of precompact Ramsey expansions),but also shows that there is no direct generalisation of the Neˇsetˇril-R¨odl Theo-rem to free amalgamation classes with closures (or strong embeddings). How-ever, perhaps surprisingly, we show that if closures are explicitly represented bymeans of partial functions, such a statement is true. We prove:
Theorem 1.3.
Let L be a language (involving relational symbols and partialfunctions) and let K be a free amalgamation class of L -structures. Then −→K isa Ramsey class. This yields an alternative proof of the Ramsey property for some recently-discovered Ramsey classes (such as ordered partial Steiner systems [4], bowtie-free graphs [19], bouquet-free graphs [8]) and also for new classes: most im-portantly a Ramsey expansion of the class of 2-orientations of a Hrushovskipredimension construction which is elaborated in [11] and which was one ofmain motivations for this paper.
A class
O ⊆ −→K has the ordering property (with respect to K ) if for every A ∈ K there exists B ∈ K such that every ordering −→ B ∈ O of B contains a copy of everyordering −→ A ∈ O of A . It is a well known that for every free amalgamation class K of relational structures the class −→K has ordering property. This fact follows byan application of Theorem 1.2 but can also be shown by more general methodsbased on hypergraphs of large girth [31, 28]. This shows that there are manyclasses K of relational structures for which −→K has the ordering property (withrespect to K ) but K itself is not a Ramsey class.For languages containing function symbols, the situation is more compli-cated. To see that some extra restriction on our class O is required, we notethe following example. Denote by T the class of all finite forests of trees rep-resented by a single unary function F connecting a vertex to its father. Let A
4e a structure containing two vertices a, b and F A ( a ) = b . A vertex c is a root if c / ∈ Dom( F A ). Any structure B can be ordered in increasing order accordingto the distance from a root vertex. It follows that such an ordering never con-tains the ordering of A given by a ≤ A b and consequently −→T does not have theordering property. Nevertheless, we show the following: Theorem 1.4.
Let L be a language (involving relational symbols and partialfunctions) and let K be a free amalgamation class of L -structures. Then thereexists an amalgamation class O ⊆ −→K of admissible orderings such that:1. every A ∈ K has an ordering in O ;2. O is a Ramsey class; and,3. O has the ordering property (with respect to K ). The details of the admissible orderings are technical and are described infull in Definition 3.2. The proof of Theorem 1.4 is a combination of the Ramseymethods used to show the ordering property of classes of relational structuresand the methods used to show the ordering property of classes with unaryfunctions (which is further elaborated in [11]). A partial automorphism of an L -structure A is an isomorphism f : D → E forsome D , E being substructures of A . We say that a class of finite L -structures K has the extension property for partial automorphisms ( EPPA , sometimes calledthe
Hrushovski extension property ) if whenever A ∈ K there is B ∈ K suchthat A is substructure of B and every partial automorphism of A extends to anautomorphism of B , see [18, 13, 14, 17, 37, 39]. In the following we will simplycall B with the property above an EPPA-extension of A .For relational languages, the extension property for partial automorphismsof free amalgamation classes can be derived from the following strengthening ofthe extension property for partial automorphisms: Theorem 1.5 (Hodkinson-Otto [17]) . Let L be a relational language, then forevery finite L -structure A there exists a finite clique faithful EPPA-extension B . A clique faithful EPPA-extension B is an EPPA-extension of A with the ad-ditional property that for every clique C in the Gaifman graph of B there existsan automorphism g of B such that g ( C ) ⊆ A . It is a well known fact that freeamalgamation classes can be equivalently described by forbidden embeddingsfrom a family of structures whose Gaifman graph is a clique and consequentlyTheorem 1.5 implies that every free amalgamation class of relational structureshas EPPA. 5he notion of irreducibility of a structure (given in Definition 2.1) is a naturalgeneralisation to the context of functional languages of the above notion ofa clique in a graph. We say that an EPPA-extension B of A is irreduciblesubstructure faithful if for every irreducible substructure C of B there exists anautomorphism g of B such that g ( C ) ⊆ A .Theorem 1.5 was further strengthened by Siniora and Solecki in the followingform. Theorem 1.6 (Siniora-Solecki [36]) . Let L be relational language. Then forevery finite relational L -structure A there exists a finite clique faithful and co-herent EPPA-extension B . Let X be a set and P be a family of partial bijections between subsets of X . Atriple ( f, g, h ) from P is called a coherent triple if Dom( f ) = Dom( h ) , Range( f ) =Dom( g ) , Range( g ) = Range( h ) and h = g ◦ f .Let X and Y be sets, and P and Q be families of partial bijections betweensubsets of X and between subsets of Y , respectively. A function ϕ : P → Q issaid to be a coherent map if for each coherent triple ( f, g, h ) from P , its image ϕ ( f ) , ϕ ( g ) , ϕ ( h ) in Q is coherent.An EPPA-extension B of A is coherent if every partial automorphism f extends to some ˆ f ∈ Aut( B ) with the property that the map ϕ from partialautomorphisms of A to automorphisms of B given by ϕ ( f ) = ˆ f is coherent.Our third main result is a strengthening of all of the above results to classesof structures with unary functions. The unarity is important in our construction. Theorem 1.7.
Let L be a language such that every function symbol F ∈ L isunary. Then for every finite L -structure A there exists a finite, irreducible sub-structure faithful, coherent EPPA-extension B . Consequently every free amal-gamation class K of finite L -structures has the coherent extension property forpartial automorphisms. It seems that ealier methods and results find a proper setting in the contextof structural Ramsey theory.
We now review some standard graph-theoretic and model-theoretic notions (seee.g. [15]).A homomorphism f : A → B is a mapping f : A → B such that for every R ∈ L R and F ∈ L F we have:(a) ( x , x , . . . , x a ( R ) ) ∈ R A = ⇒ ( f ( x ) , f ( x ) , . . . , f ( x a ( R ) )) ∈ R B , and,(b) f (Dom( F A )) ⊆ Dom( F B ) and f ( F A ( x , x , . . . , x d ( F ) )) = F B ( f ( x ) , f ( x ) , . . . , f ( x d ( F ) )) . B CB α α β β Figure 1: An amalgamation of B and B over A .For a subset A (cid:48) ⊆ A we denote by f ( A (cid:48) ) the set { f ( x ) : x ∈ A (cid:48) } and by f ( A )the homomorphic image of a structure A .If f is injective, then f is called a monomorphism . A monomorphism f isan embedding if for every R ∈ L R and F ∈ L F :(a) ( x , x , . . . , x a ( R ) ) ∈ R A ⇐⇒ ( f ( x ) , f ( x ) , . . . , f ( x a ( R ) )) ∈ R B , and,(b) for every F ∈ L F it holds that( x , . . . , x d ( F ) ) ∈ Dom( F A ) ⇐⇒ ( f ( x ) , . . . , f ( x d ( F ) )) ∈ Dom( F B ) . If f is an embedding which is an inclusion then A is a substructure (or subobject )of B . For an embedding f : A → B we say that A is isomorphic to f ( A ) and f ( A ) is also called a copy of A in B . Thus (cid:0) BA (cid:1) is defined as the set of all copiesof A in B .Given A ∈ K and B ⊂ A , the closure of B in A , denoted by Cl A ( B ), is thesmallest substructure of A containing B . Closure in A is unary if Cl A ( B ) = (cid:83) v ∈ B Cl A ( v ) for all B ⊂ A .Let A , B and B be structures, α an embedding of A into B and α anembedding of A into B . Then every structure C with embeddings β : B → C and β : B → C such that β ◦ α = β ◦ α is called an amalgamation of B and B over A with respect to α and α . See Figure 1. We will call C simplyan amalgamation of B and B over A (as in most cases α and α can bechosen to be inclusion embeddings). We say that such an amalgamation is free if β ( x ) = β ( x ) if and only if x ∈ α ( A ) and x ∈ α ( A ) and there are notuples in any relations of C and no tuples in Dom( F C ), F ∈ L F , using verticesof both β ( B \ α ( A )) and β ( B \ α ( A )). Definition 1.1. An amalgamation class is a class K of finite structures satis-fying the following three conditions:1. Hereditary property:
For every A ∈ K and a substructure B of A we have B ∈ K ; 7. Joint embedding property:
For every A , B ∈ K there exists C ∈ K suchthat C contains both A and B as substructures;3. Amalgamation property:
For A , B , B ∈ K and α embedding of A into B , α embedding of A into B , there is C ∈ K which is an amalgamationof B and B over A with respect to α and α .If the C in the amalgamation property can always be chosen as the free amal-gamation, then K is a free amalgamation class .We will give examples of free amalgamation classes (in the case where thelanguage L is not relational) in Section 5. The proof of Theorem 1.3 is a variation of the
Partite Construction intro-duced by Neˇsetˇril and R¨odl for classes of hypergraphs and relational structures(see [30]) which was recently extended to classes with unary [19] and later gen-eral closures [20, 4]. The Partite Construction is a machinery which allows oneto transform one Ramsey class into another, more special, Ramsey class. Whatfollows is a partite construction proof (as used previously, for example, in [26])done in the context of structures involving partial functions.The basic Ramsey class in our construction will be provided by the follow-ing result about the Ramsey property of ordered structures with relations and(total) functions.
Theorem 2.1 (Ramsey theorem for finite models, Theorem 4.26 of [20]) . Sup-pose L is a language containing a relational symbol ≤ and let −→ Str( L ) be theclass of all finite L -structures where ≤ is a linear ordering of the vertices andall functions are total. Then −→ Str( L ) is a Ramsey class. This is a strengthening of the theorem giving the Ramsey property of orderedrelational structures proved independently by Neˇsetˇril-R¨odl [32] and Abramson-Harrington [1] in 1970s. Considering functions is a rather difficult task and theproof of Theorem 2.1 involves a recursive nesting of the Partite Constructions toestablish valid non-unary closures. Building on Theorem 2.1 our task is signifi-cantly easier and we only concentrate on further refining the Ramsey structuregiven by Theorem 2.1 into one belonging to a given free amalgamation class.This is done by tracking all irreducible substructures of the object constructed.A relational structure A is irreducible if every pair of vertices belongs to sometuple in a relation of A . It is well known that every free amalgamation class K of relational structures can be equivalently described as a class of finite struc-tures that contains no copies of structures from a fixed family F of irreduciblerelational structures. In fact the family F consists of all minimal structuresnot belonging to class K . This easy observation explains the correspondence ofTheorem 1.1 and Theorem 1.2. 8 bc d ( a, b ) ∈ RF ( a ) = c F ( b ) = d Figure 2: An example of an irreducible structure with a binary relation R anda unary function F .Our construction is based on the following refinement of the notion of ir-reducible structure in the context of structures with partial functions (whichallows us to strengthen the construction in [20]): Definition 2.1. An L -structure A is irreducible if it cannot be created as afree amalgamation of any two of its proper substructures. Example.
Consider the language L consisting of one binary relation R andone unary function F . An example of an irreducible structure is a structure A (depicted in Figure 2) on vertices A = { a, b, c, d } where ( a, b ) ∈ R A , Dom( F A ) = { a, b } and F A ( a ) = { c } , F A ( b ) = { d } . This structure is reducible if F is seenas a relation rather than function.The basic part of our construction of Ramsey objects is a variant of thePartite Lemma (introduced in [30] and refined for closures in [20]) which dealswith the following objects. Definition 2.2 ( A -partite system) . Let L be a language and A an L -structure.Assume A = { , , . . . , a } . An A -partite L -system is a tuple ( A , X B , B ) where B is an L -structure and X B = { X B , X B , . . . , X a B } is a partition of the vertexset of B into a many classes X i B , called parts of B , such that1. the mapping π which maps every x ∈ X i B to i , i = 1 , , . . . , a , is a homo-morphism B → A ( π is called the projection );2. every tuple in every relation of B meets every class X i B in at most oneelement (i.e. these tuples are called transversal with respect to the parti-tion).3. for every function F ∈ L F it holds that for every (cid:126)t ∈ Dom( F B ) the tuplecreated by concatenation of (cid:126)t and F B ( (cid:126)t ) (in any order) is transversal.The isomorphisms and embeddings of A -partite systems, say of B into B ,are defined as the isomorphisms and embeddings of structures together with thecondition that all parts are being preserved (the part X i B is mapped to X i B for every i = 1 , , . . . , a ). Of course, A itself can be considered as an A -partitesystem.We say that an A -partite L -system is transversal if all of its parts consist ofat most one vertex. 9 emma 2.2 (Partite Lemma with relations and functions) . Let L be a language, A be a finite L -structure, and B be a finite A -partite L -system. Then thereexists a finite A -partite L -system C such that C −→ ( B ) A . Moreover if every irreducible subsystem of B is transversal, then we can alsoensure that every irreducible subsystem of C is transversal. (Compare the Partite Lemma in [20]. Here we newly introduced the state-ment about transversality of irreducible substructures. Note that the embed-dings considered in the Ramsey statement are all as A -partite systems.)Advancing the proof of Lemma 2.2, for completeness, we briefly recall theHales-Jewett Theorem [12]. Consider the family of functions f : { , , . . . , N } → Σ for some finite alphabet Σ. A combinatorial line L is a pair ( ω, h ) where ∅ (cid:54) = ω ⊆ { , , . . . , N } and h is a function from { , , . . . , N } \ ω to Σ. The com-binatorial line L describes the family of all those functions f : { , , . . . , N } → Σthat are constant on ω and f ( i ) = h ( i ) otherwise. The Hales-Jewett Theoremguarantees, for sufficiently large N , that for every 2-colouring of the functions f : { , , . . . , N } → Σ there exists a monochromatic combinatorial line.
Proof of Lemma 2.2.
Assume without loss of generality A = { , , . . . , a } anddenote by X B = { X B , X B , . . . , X a B } the parts of B . We take N sufficiently large(that will be specified later) and construct an A -partite L -system C with parts X C = { X C , X C , . . . , X a C } as follows:1. For every 1 ≤ i ≤ a let X i C be the set of all functions f : { , , . . . , N } → X i B .
2. For every relation R ∈ L R , put( f , f , . . . , f a ( R ) ) ∈ R C if and only if for every 1 ≤ i ≤ N it holds that( f ( i ) , f ( i ) , . . . , f a ( R ) ( i )) ∈ R B .
3. For every function F ∈ L F , put F C ( f , f . . . , f d ( F ) )( i ) = F B ( f ( i ) , f ( i ) , . . . , f d ( F ) ( i ))if and only if for every j = 1 , , . . . , N ( f ( j ) , f ( j ) , . . . , f d ( F ) ( j )) ∈ Dom( F B ) . C . It is easy to check that C is indeed an A -partite L -system with parts X C = { X C , X C , . . . , X a C } .We verify that, if N is large enough, C −→ ( B ) A . Let (cid:101) A , (cid:101) A , . . . , (cid:101) A t be an enumeration of all subsystems of B which are isomorphic to A . PutΣ = { , , . . . , t } which we consider as an alphabet. Each combinatorial line L = ( ω, h ) in Σ N corresponds to an embedding e L : B → C which assigns toevery vertex v ∈ X p B a function e L ( v ) : { , , . . . , N } → X p B (i.e. a vertex of X p C ) such that: e L ( v )( i ) = (cid:40) v for i ∈ ω , and,the unique vertex in (cid:101) A h ( i ) ∩ X p B otherwise.It follows from the construction of C and from the fact that B has a projectionto A that e L is an embedding.Let N be the Hales-Jewett number guaranteeing a monochromatic line inany 2-colouring of the N -dimensional cube over an alphabet Σ. Now assumethat A , A is a 2-colouring of all copies of A in C . Using the construc-tion of C we see that among copies of A are copies induced by an N -tuple( (cid:101) A u (1) , (cid:101) A u (2) , . . . , (cid:101) A u ( N ) ) of copies of A in B for every function u : { , , . . . ,N } → { , , . . . , t } . However such copies are coded by the elements of the cube { , , . . . , t } N and thus there is a monochromatic combinatorial line L . Themonochromatic copy of B is then e L ( B ).Finally we verify that if every irreducible subsystem of B is transversal thenalso every irreducible subsytem of C is transversal. Assume the contrary and de-note by D a non-transversal irreducible subsystem of C . Denote by f , f , . . . , f n an enumeration of all distinct vertices of D . By non-transversality assume that f and f are in the same part. For every i ∈ , , . . . , N denote by D i the sub-structure of B on vertices f ( i ) , f ( i ) , . . . , f n ( i ). Because D i is a homomorphicimage of D it is irreducible and thus it follows that D i is transversal. Conse-quently f ( i ) = f ( i ). Because this holds for every choice of i , we have f = f .A contradiction. Proof of Theorem 1.3.
Given A , B ∈ −→K ⊆ −→ Str( −→ L ) use Theorem 2.1 to obtain C −→ ( B ) A . This is clearly possible when all functions in B are total. In case they are not, itis possible to extend the language L by new relations that represent the domainof each partial function and turn every symmetric partial function F ∈ L F toa complete function by defining F B ( (cid:126)t ) = (cid:126)u , where (cid:126)u is a tuple consisting of theminimal vertex of (cid:126)t (in the order ≤ B ) repeated r ( F ) times, and by orderingtuples using ≤ . Such a ‘completion’ makes every function total and preserve allcopies of A and B . Once Theorem 2.1 has been applied, we pass back to theoriginal language and remove what was added in the ‘completion’.11 P (cid:101) B (cid:101) B (cid:101) B (cid:101) B (cid:101) B (cid:101) B Figure 3: The construction of P .Enumerate all copies of A in C as { (cid:101) A , (cid:101) A , . . . , (cid:101) A b } . We will define C -partite systems (‘pictures’) P , P , . . . , P b such that(i) every irreducible subsystem E of P k is transversal and (if seen as a struc-ture) is isomorphic to some substructure of B , and,(ii) in any 2-colouring of (cid:0) P k A (cid:1) , 1 ≤ k ≤ b , there exists a copy (cid:101) P k − of P k − such that all copies of A with a projection to (cid:101) A k are monochromatic.We then show that putting C to be P b (seen as a structure) with the linearorder completed arbitrarily (extending the order of the parts), we have thedesired Ramsey property C −→ ( B ) A .We first verify that if (i) holds, then C ∈ −→K . Assume the contrary. Denoteby F the minimal substructure of C such that F / ∈ −→K . Because K is a freeamalgamation class, we know that the unordered reduct of F is irreducible. By(i) however F is a substructure of B ∈ K . A contradiction to K being hereditary.It remains to prove (i) and (ii). Put C = { , . . . , c } and X P k = { X k , X k ,. . . , X ck } . We proceed by induction on k .1. The picture P is constructed as a disjoint union of copies of B . For everycopy (cid:101) B of B in C we include a copy (cid:101) B (cid:48) of (cid:101) B in P which projects onto (cid:101) B . The copies corresponding to different (cid:101) B are disjoint (see Figure 3).This indeed satisfies (i).2. Suppose the picture P k is already constructed. Let B k be the substruc-ture of P k induced by P k on vertices which project to (cid:101) A k +1 . We usethe Partite Lemma 2.2 to obtain an (cid:101) A k +1 -partite system D k +1 with D k +1 −→ ( B k ) (cid:101) A k +1 . Now consider all copies of B k in D k +1 and ex-tend each of these structures to a copy of P k (using free amalgamation of C -partite systems). These copies are disjoint outside D k +1 . In this ex-tension we preserve the parts of all the copies. The result of this multipleamalgamation is P k +1 . Because D k +1 −→ ( B k ) (cid:101) A k +1 we know that P k +1 satisfies (ii).From the ‘Moreover’ part of Lemma 2.2, we may assume that D k +1 sat-isfies (i). Because P k +1 is created by a series of free amalgamations, itfollows that P k +1 also satisfies (i).12ut C = P b . It follows easily that C −→ ( B ) A . Indeed, by a backwardinduction on k one proves that in any 2-colouring of (cid:0) CA (cid:1) there exists a copy (cid:101) P of P such that the colour of a copy of A in (cid:101) P depends only on its projectionto C . As this in turn induces a colouring of the copies of A in C , we obtaina monochromatic copy of B in (cid:101) P . Remark.
This proof of Theorem 1.3 follows the Partite Construction developedin [29, 30]. It may be also seen as a cleaner proof of a particular case of themore general (and more elaborate) Theorem 2.2 of [20].
For many Ramsey classes K , the class −→K of free orderings of structures in K hasthe ordering property. In fact, for free amalgamation classes we can give a fullcharacterisation by means of the following easy (and folklore) proposition. Proposition 3.1.
Let K be a free amalgamation class of L -structures. Then −→K has the ordering property if and only if all closures Cl A ( u ) of vertices aremutually isomorphic single element structures. Remark.
It does not follow from Proposition 3.1 that the class K must haveonly trivial closures in order for −→K to have the ordering property. Consider forexample a class of structures with one binary function F such that the domain of F does not contain tuples with duplicated vertices. Here all vertices are closed,but pairs of vertices have non-trivial closures. A related example is discussed inSection 5.2. Proof.
Assume that there is −→ A ∈ K and u, v ∈ −→ A such that Cl A ( u ) is notisomorphic to Cl A ( v ). Then one can choose an ordering −→ A of A so that the setof all vertices v (cid:48) ∈ A such that Cl A ( v (cid:48) ) is isomorphic to Cl A ( v ) forms an initialsegment. Now assume, to the contrary, that there exists B ∈ K such that everyordering of B contains a copy of −→ A . This is clearly not possible because onecan choose an ordering of B such that all vertices u (cid:48) ∈ B such that Cl A ( u (cid:48) ) isisomorphic to Cl A ( u ) forms an initial segment. This is a contradiction to −→K having the ordering property.Now consider the case that all closures of vertices are isomorphic, but nottrivial. Because the intersection of two closures is also closed, it follows thatclosures of vertices are disjoint and thus it is always possible to order structuresin a way that all vertex closures form intervals. It follows that there is no B witnessing the ordering property for any −→ A ∈ K which is ordered so that somevertex closure is not an interval.Finally assume that K is a free amalgamation class where all closures ofvertices are trivial and mutually isomorphic. We may assume that there are nounary relations. Given −→ A ∈ −→ K we construct −→ B from a disjoint copy of −→ A and ←− A (by this we mean a structure created from −→ A by reversing the linear ordering13f vertices). Now extend −→ B to −→ B by adding, between every neighbouringpair of vertices u ≤ −→ B v ∈ B a new vertex n u,v . Extend the order so u ≤ −→ B n u,v ≤ −→ B v . By free amalgamation, −→ B ∈ −→K .Denote by −→ I ∈ −→K a structure consisting of two vertices u ≤ −→ I v and norelations containing both of them besides ≤ −→ I . Find −→ B ∈ −→K with −→ B −→ ( −→ B ) −→ I by application of Theorem 1.3.We verify that −→ B has the desired property. Let −→ C be any re-ordering of −→ B .This re-ordering induces a coloring of (cid:0) −→ B −→ I (cid:1) : if the order of the points in some −→ I (cid:48) ∈ (cid:0) −→ B −→ I (cid:1) agrees with thier order in −→ C , then color −→ I (cid:48) red, and blue otherwise.The monochromatic copy of −→ B will have the property that it is either orderedin the same way as −→ B or the order is reversed. By construction of −→ B it followsthat in both alternatives there is a copy of −→ A .In the following we generalize the main idea of this proof (the idea of whichgoes back to [31]) to classes with non-trivial closures of vertices. Free orderingsdo not suffice anymore and we have to define carefully the admissible orderings. Definition 3.1.
Let A be an L -structure. If a, b ∈ A we write a ∼ A b ifCl A ( a ) = Cl A ( b ). This is an equivalence relation on A and we refer to theclasses as the closure-components of A . The class containing a will be denotedby Cc A ( a ).If a ∈ A we define the level l A ( a ) of a in A inductively. We say that l A ( a ) = 0in the case where Cc A ( a ) = Cl A ( a ); otherwise l A ( a ) = l A ( b ) + 1 where b is avertex of the maximal level in A amongst vertices in Cl A ( a ) \ Cc A ( a ).We say that A ∈ K is a closure-extension at level k if there is a uniqueclosure-component C of vertices of level k in A and Cl A ( a ) = A for every a ∈ C . In this case, we write A ◦ = A \ C . Every closure of a vertex is aclosure-extension.We say that two closure-components C and C (cid:48) of −→ A (or their closures) are homologous if Cl A ( C ) and Cl A ( C (cid:48) ) are isomorphic and Cl A ( C ) \ C = Cl A ( C (cid:48) ) \ C (cid:48) . Note that the isomorphism must fix each vertex of Cl A ( C ) \ C and, if C (cid:54) = C (cid:48) , then Cl A ( C ) \ C = Cl A ( C ) ∩ Cl A ( C (cid:48) ) is closed in A . Example.
Consider the class T of forests represented by a single unary function(the predecessor relation) described in Section 1.2. In this class the closure ofa vertex is the path to a root vertex. Every vertex thus forms a trivial closurecomponent and its level is determined by the distance to the root vertex. In thisparticular case a structure is a closure-extension if and only if A is an orientedpath (that is structure on vertex set A = { v , v , . . . , v n } and F A ( v i ) = v i +1 forevery 1 ≤ i < n ). 14 a bF ( c ) = { a, b } F ( a, b ) = { c } Figure 4: Example of an closure-extension A where A ◦ is not a substructure.Observe also that for a closure-extension A (depicted in Figure 4), the setof vertices A ◦ is not necessarily closed. Consider a language with a function F from 1-tuples to sets of two vertices and a function F from 2-tuples tosingletons. The structure A with A = { a, b, c } , F A ( c ) = { a, b } , F A ( a, b ) = { c } is an closure-extension of level 1: l ( a ) = l ( b ) = 0 and l ( c ) = 1, the set A ◦ is { a, b } , however Cl A { a, b } = { a, b, c } .Suppose A , B ∈ K are closure-extensions and −→ A , −→ B ∈ −→K are orderings. Wesay that these are similar if there is an isomorphism α : A → B which is alsoorder-preserving when seen as a mapping α : A ◦ → B ◦ . This is an equivalencerelation and in our admissible orderings, we choose a fixed representative fromeach similarity-type of ordered closure-extension.In what follows we shall assume that we have fixed some total ordering −→ A (cid:69) −→ B between isomorphism types of orderings of closure-extensions such thatS1 | A | < | B | implies −→ A (cid:69) −→ B .(In particular, (cid:69) is a well ordering.)First we define a preorder of vertices which we will later refine to a linearorder. Given two vertices u (cid:54) = v ∈ −→ A we write u (cid:52) −→ A v if one of the followingholds:P1 Cl −→ A ( u ) (cid:69) Cl −→ A ( v ) and they are not isomorphic;P2 Cl −→ A ( u ) is isomorphic to Cl −→ A ( v ) but Cl −→ A ( u ) \ Cc −→ A ( v ) is lexicographicallybefore Cl −→ A ( v ) \ Cc −→ A ( v ) considering the order ≤ −→ A ;P3 Cl −→ A ( u ) and Cl −→ A ( v ) are homologous closure-extensions.Note that this is indeed a preorder on the vertices of A . We can now describeour class of admissible orderings. Definition 3.2.
Suppose K is a free amalgamation class. We say that O ⊆ −→K is a class of admissible orderings of structures in K if the following conditionshold.A1 If A ∈ K , then there is some ordering ≤ −→ A of A in O .152 O is closed for substructures.A3 For every −→ A ∈ O , the ordering ≤ −→ A refines (cid:52) −→ A .A4 For every −→ A ∈ O , the closure-components form linear intervals in ≤ −→ A .A5 For every B ∈ K , if A , A , . . . , A n is a family of substructures and ≤ isa linear order of A = ∪ ≤ i ≤ n A i such that(a) ≤ satisfies the conclusions of A3 and A4; and(b) each substructure of B contained in A is admissibly ordered by ≤ ;then there exists −→ B ∈ O such that ≤ −→ B restricted to A is ≤ .A6 Suppose that −→ A , −→ B ∈ O are similar ordered closure-extensions. Then −→ A is isomorphic to −→ B . Example.
Consider A ∈ T where T is the class of forests discussed in Sec-tion 1.2. Because the closure-extensions are all formed by oriented paths, theorder (cid:69) requires vertices to be ordered according to their levels (in particularall root vertices come first and can be ordered arbitrarily). The sons of a vertex v ∈ A are trivial homologous components and thus they are required to alwaysform an interval and the order amongst these intervals is given by order of theirfathers.There is some flexibility in the definition of admisibility. For example it ispossible to order forests in a way that every tree (and recursively every subtree)forms an interval. The particular choice is however not very important as it canbe shown that they are all equivalent up to bi-definability (this follows as O isa Ramsey class, see [22]). Proposition 3.2.
Suppose K is a free amalgamation class. Then there is aclass O of admissible orderings of K .Proof. We proceed by induction on −→K ordered abitrarily in order of increasingnumber of vertices. In this order, for every −→ C ∈ −→K we decide if −→ C ∈ O ornot by a variant of a greedy algorithm. In the induction step assume that wealready decided the presence in O for every proper substructure of −→ C .We put −→ C ∈ O if and only if:O1 −→ C satisfies A3 and A4,O2 every proper substructure of −→ C is in O , and,O3 if −→ C is an ordered closure-extension, then there is no similar but non-isomorphic −→ D ∈ O .This finishes the description of O . We verify that the conditions of Definition 3.2are satisfied. 16irst we check A5. Assume, to the contrary, that there are B ∈ K , substruc-tures A , A , . . . , A n and a linear order ≤ on A without such an extension.From all counter-examples choose one minimizing | B | and among those, mini-mize | B |−| A | .We consider three cases:1. Suppose B is not a closure-extension and B = A . It follows that theorder ≤ satisfies both O1 and O2 and thus we can put ≤ −→ B = ≤ and obtain −→ B ∈ O , a contradiction.2. Suppose B is a closure-extension and B ◦ = A . Extend ≤ to an order of −→ B in a way it satisfies O1. In this case −→ B satisfies O2 because no propersubstructure contains B \ B ◦ . Furthermore, we may assume that O3 holds.This is a contradiction to −→ B / ∈ O .3. Suppose neither of the previous cases apply. Then there is a properclosure-extension C ⊆ B such that C (cid:54)⊆ A . Denote by ≤ (cid:48) the order ≤ restricted to C ∩ A and consider the structures C i = A i ∩ C , 1 ≤ i ≤ n .From the minimality of the counter-example and because | C | < | B | weknow that order of ≤ (cid:48) can be extended to an admissible order −→ C of C .Now extend to C ∪ A the orders ≤ and ≤ −→ C in such a way that A3 and A4are satisfied. This combined order along with the family of substrucutres C , A , A , . . . , A n contradicts the second assumption about the minimal-ity of the counter-example.This finishes proof of A5.Condition A1 is implied by A5. Conditions A2, A3, A4 and A6 followsdirectly from the construction of O . Advancing the proof of Theorem 1.4 we show two lemmas which generalize themain ideas of proof of Proposition 3.1.
Lemma 3.3.
Let R be a Ramsey class of ordered structures. Then for every −→ A ∈ R there exists −→ B ∈ R such that every re-ordering −→ C ∈ R of −→ B contains are-ordering (cid:101) A of −→ A such that every two isomorphic substructures of −→ A are alsoisomorphic in the order of (cid:101) A .Proof. Let −→ A be a structure and −→ A a substructure. Denote by n the number ofpossible re-orderings of −→ A (i.e. n = | A | !). By the Ramsey property construct −→ B such that −→ B −→ ( −→ A ) −→ A n . Every re-ordering of −→ B induces an n -coloringof copies of −→ A in −→ B and so by the Ramsey property there exists a copy (cid:101) A of re-ordered −→ A in −→ B having the property that all copies of −→ A in (cid:101) A areordered the same way. The statement follows by iterating this argument forevery substructure of −→ A . 17 C C C CC C C C C C C C C −→ A −→ A ←− A −→ A C C −→ I Figure 5: Construction used in the proof of Lemma 3.4. Pairs of closure com-ponents which are not necessarily free are connected by dotted lines.
Lemma 3.4.
Let K by a free amalgamation class. Then for every −→ A ∈ −→K in which every closure-component forms an interval (in the order of −→ A ) thereexists −→ B such that every re-ordering −→ C of −→ B where every closure-componentforms an interval (in the order of −→ C ) contains a copy of a re-ordering of −→ A where the order between vertices in distinct homologous closure-components ispreserved.Proof. For simplicity we show the construction for a given −→ A and two distincthomologous closure-components C ≤ −→ A C . The full statement can be shownby iterating the argument for every such pair. The construction is schematicallydepicted in Figure 5.First observe that C and C are not in the closure of Cl A ( C ) \ C =Cl A ( C ) \ C because this would imply C = C .Now, by a free amalgamation construct −→ A extending −→ A with a new closure-component C homologous to both C and C where the linear order is extendedso that C ≤ −→ A C ≤ −→ A C . Denote by ←− A the structure created from −→ A by re-ordering closure-components C , C and C so that C ≤ ←− A C ≤ ←− A C . Denoteby −→ A the free amalgamation of −→ A and ←− A over Cl A ( C ) \ C . Because K isa free amalgamation class we have −→ A , ←− A , −→ A ∈ −→K .Note that the addition of C into A is necessary only when Cl( −→ A ( C ∪ C ))is not a result of free amalgamation of two copies of Cl −→ A ( C ) . In general theremay be relations spanning C and C which would make use of the Ramseyargument bellow impossible.Put −→ I = Cl −→ A ( C ∪ C ) and use Theorem 1.3 to find −→ B ∈ −→K containing −→ A such that −→ B −→ ( −→ A ) −→ I . Now every re-ordering −→ C of −→ B induces a 2-coloringof the copies of −→ I in −→ B which in turn leads to the existence of a copy of are-ordering of −→ A where the order of C and C is preserved.Now we are finally ready to prove the main result of this section.18 roof of Theorem 1.4. Given −→ A ∈ O we construct −→ B ∈ −→K with −→ A as a sub-structure such that every admissible re-ordering −→ C ∈ O of −→ B contains a copyof −→ A .By application of Lemmas 3.4 and 3.3 it is enough to construct an admissiblyordered −→ B such that that every re-ordering −→ C ∈ O of −→ B with the followingtwo properties contains a copy of −→ A :(a) the order of vertices in distinct homologous components is preserved;(b) the order in −→ C on substructures which are closures of vertices dependsonly on their isomorphism type in −→ B .Given −→ A denote by −→ A , −→ A ,. . . , −→ A n all admissible re-orderings of −→ A havingproperties (a) and (b). So these structures all have the same domain, and onlydiffer in their orderings. Put −→ B to be the disjoint union of −→ A , −→ A ,. . . , −→ A n with the order completed arbitrarily. Let −→ C be an admissible reordering of −→ B satisfying (a) and (b). Denote by α i : −→ A → −→ C the map which sends a ∈ A to a ∈ A i , for i ≤ n . It is enough to prove the following. Claim:
For all k ≥
0, there is some i ≤ n such that the map α i preserves theordering on vertices in −→ A of level at most k .We prove this by induction on k . Denote by A | k the set of vertices in A atlevel ≤ k (this is not necessarily a substructure or even a structure in K , but itstill makes sense to discuss admissible orderings of it as it contains the closuresof all of its vertices, and the criteria for being an admissible ordering dependonly on these). By setting A | − = ∅ , we can incorporate the proof for the basecase k = 0 into the general argument. Step 1:
Every admissible ordering of A | k − satisfying (a) and (b) extends to oneof A | k . If X is the closure of a vertex of level k in A , then any two such closure-extensions differ on X by a permutation in Aut( X /X ◦ ) (that is, automorphismsof X fixing all vertices of level less than k ).Indeed, we need only say how to define the ordering on X . But, given theordering on X ◦ , this is determined by condition A6 in Definition 3.2, up to theaction of Aut( X /X ◦ ). It is easy to see that the resulting ordering on A | k isadmissible. Step 2:
Suppose the claim holds up to level k −
1. Let I denote the set of i ≤ n for which α i restricted to A | k − is order-preserving. So this is non-empty. Let i ∈ I and let X be as in Step 1.There is β i ∈ Aut( X /X ◦ ) such that α i ◦ β i preserves the ordering on X andso is an isomorphism between −→ X , the structure on X in −→ A and the substructure α i ( X ) in −→ C . By Step 1, there is some j ≤ n such that the map β − i , regardedas a map from −→ A i to −→ A j (in −→ B ) is order-preserving and all vertices in A | k − have the same ordering in −→ A i and −→ A j . By condition (b), it follows that this mapbetween the corresponding subsets of −→ C is order-preserving (as all orderings19re determined by what happens in closures of vertices). Thus j ∈ I and α j isorder-preserving on X . Repeating this argument for other vertices at level k ,we complete the inductive step.We have verified that the class of admissible orderings O has the order-ing property. The Ramsey property follows from Theorem 1.3 and A5: Given −→ A , −→ B ∈ O and −→ C ∈ −→K such that −→ C −→ ( −→ B ) −→ A . Construct a new order ≤ of −→ C in a way that ≤ agrees with ≤ C on every copy of −→ B in −→ C . Complete ≤ so itsatsifies A3 and A4. By A5 it follows that C ordered by ≤ is in O . The proof of Theorem 1.7 is a variant of the proof of clique-faithful EPPA byHodkinson and Otto [17] combined with a proof of EPPA for purely functionallanguage from [11]. As in [17] our starting point is the following constructiongiving EPPA for relational structures and we verify coherency as in [35, 36].
Theorem 4.1 (Herwig [13], coherency verified by Solecki [38]) . For any rela-tional language L , the class of all finite L -structures has coherent EPPA. To apply this construction to structures with functions we will temporarilyinterpret functions as relational symbols.
Definition 4.1.
Suppose L is a language where all function symbols are unary.Given an L -structure A we denote by A − its relational reduct constructed asfollows. The language L − of A − is a relational language containing all relationalsymbols of A and additionally containing for every function symbol F ∈ L F a relation symbol R F ∈ L −F of arity a ( R F ) = d ( F ) + 1. The vertex set of A is the same as the vertex set of A − . For every R ∈ L R we have R A = R A − and for every F ∈ L F it holds that ( t , t , . . . , t d ( F ) , t d ( F )+1 ) ∈ R F A − if andonly if F A ( t , t , . . . , t d ( F ) ) = { t d ( F )+1 ) } (recall: our functions assign subsets totuples).Note that in the above, the structures A and A − have the same automor-phisms and any partial automorphism of A is an automorphism of A − (but ofcourse not conversely).The proof of Theorem 1.7 will occupy the rest of this section. Proof of Theorem 1.7.
Given A we invoke Theorem 4.1 to obtain a coherentEPPA-extension B − of A − (with respect to all partial isomorphisms of A − ). Weuse the following terminology, following an exposition by Hodkinson in [16, 17].The difference between our proof and that in [17] and [36] (which give cliquefaitful EPPA for structures in relational lanugages) is that whereas in thesepapers, the vertices in the EPPA-extension are of the form ( v, χ v ) (as in thenotation below), our vertices will actually be sets of such vertices (carrying an L -structure). 20 set S ⊆ B − is called small if there is some g ∈ Aut( B − ) such that g ( S ) ⊆ A . Otherwise S is called big . Denote by U the set of all big subsets of B − and note that this is preserved by Aut( B − ). Given b ∈ B − a map χ : U → N is a b -valuation function if χ ( S ) = 0 for all b / ∈ S ∈ U and 1 ≤ χ ( S ) < | S | otherwise.Given vertices a, b ∈ B − and their valuation functions, χ a and χ b we saythat pairs ( a, χ a ) and ( b, χ b ) are generic if either ( a, χ a ) = ( b, χ b ), or a (cid:54) = b andfor every S ∈ U such that a, b ∈ S it holds that χ a ( S ) (cid:54) = χ b ( S ).The key construction in the proof is the following “local covering” con-struction of a b -valuation L -structure V b . Fix b ∈ B − and an automorphism α : B − → B − such that α ( b ) ∈ A − (we can assume such an automorphismalways exists — all other vertices can be removed from B − ). Now consider the L -substructure V αb = Cl A ( α ( b )) of A . Suppose that for every v ∈ α − ( V αb ) wehave a v -valuation function χ v such that the assigned valuation functions aregeneric for every pair of vertices in α − ( V αb ). (Such a choice of valuation func-tions always exists and we will show how to obtain it later when we define anembedding φ of A .) Denote by V b the set of all such pairs ( v, χ v ), v ∈ α − ( V αb ).On the set V b we consider the L -structure V b , called a b -valuation , which is de-fined in such a way that the composition of mappings α and π ( v, χ v ) = v formsan embedding α ◦ π : V b → V αb . (This is a standard construction, we use the1–1 mapping α ◦ π to pull back the structure V αb to V b ; note that the structurehere does not depend on the choice of α .) Observe that then Cl V b (( b, χ b )) = V b .Notice that for every b there are multiple choices of b -valuations V b (de-pending on particular choice of valuation functions assigned to vertices, but notdepending on the choice of α ). The sets V b and structures V b will form a “coverof B − ” and we find it convenient to make the following definitions. Definition 4.2.
Recalling that all functions of L are unary, we say that a pairof valuations V a and V b is generic if(i) every pair of vertices ( u, χ u ) ∈ V a and ( v, χ v ) ∈ V b is generic;(ii) for every ( u, χ u ) ∈ V a and ( v, χ v ) ∈ V b and F ∈ L F it holds that(a) if ( u, v ) ∈ R F B − , then ( v, χ v ) ∈ V a and(b) if ( v, u ) ∈ R F B − , then ( u, χ u ) ∈ V b ;(iii) if ( u, χ u ) ∈ V a ∩ V b , then Cl V a (( u, χ u )) = Cl V b (( u, χ u )).We also say that a set S of valuations is generic if every pair of valuations in S is generic.Now we construct an L -structure C :1. The vertices of C are all b -valuation L -structures V b , for b ∈ B − .2. For every relation R ∈ L F put ( V v , V v , . . . , V v a ( R ) ) ∈ R C if and only if( v , v , . . . , v a ( R ) ) ∈ R B − and the set { V v i : 1 ≤ i ≤ a ( R ) } is generic.21. For every function F ∈ L F put F C ( V v ) = { V v , V v , . . . , V v r ( F )+1 } ifand only if ( v , v l ) ∈ R F B − for every 2 ≤ l ≤ r ( F ) + 1 and the set { V v i :1 ≤ i ≤ r ( F ) + 1 } is generic.First we verify that C is indeed an L -structure. It will suffice to show thatif F ∈ L F , V v ∈ C and F ( v ) = { u , . . . , u s } (where s = r ( F )), then thereare unique V u , . . . , V u s ∈ C with { V v , V u , . . . , V u s } generic (and therefore F C ( V v ) = { V u , . . . , V u s } ). But as ( v, u i ) ∈ R F B − , genericity of V v , V u i impliesthat V u i ⊆ V v . So V u i = Cl V v ( u i , χ u i ), where ( u i , χ u i ) ∈ V v .Next we give an embedding φ : A → C with generic image. For every big set S ∈ U choose f S : S → { , , , . . . , | S | − } to be a function such that f S ( v ) > v ∈ A ∩ S and for every pair of vertices u, v ∈ A ∩ S it holds that f S ( u ) (cid:54) = f S ( v ). Such a function exists because A ∩ S is always a proper subsetof S . Given a vertex a ∈ A we put φ ( a ) to be an a -valuation constructed fromCl A ( a ) by mapping every vertex v ∈ Cl A ( a ) to ( v, χ v ) where χ v ( S ) = f S ( v ).It is easy to verify that this is indeed an embedding from A to C and φ ( A ) isgeneric.We aim to show that C is an EPPA-extension of φ ( A ). We first take time toprove a lemma which will allow us to use the fact that B − is an EPPA-extensionof A − . Denote by V the union of all vertex sets of V v ∈ C . If g ∈ Aut( B − ),we say that the partial map q : V → V is g -compatible if for all ( a, χ ) ∈ Dom( q )there exists a g ( a )-valuation function χ (cid:48) such that q (( a, χ )) = ( g ( a ) , χ (cid:48) ). Let g ∈ Aut( B − ) and p : C → C be a partial automorphism. We say that p is g -compatible if there exists a g -compatible map q : V → V such that forall V v ∈ Dom( p ) q is an isomorphism of V v and p ( V v ). Denote by π thehomomorphism (projection) C − → B − defined by π ( V v ) = v . Lemma 4.2.
Let p : C → C be a partial automorphism with generic domainand range, g ∈ Aut( B − ) , and suppose that p is g -compatible. Then p extendsto some g -compatible ˆ p ∈ Aut( C ) .Proof. As Dom( p ) is generic, for every v ∈ π (Dom( p )) there is precisely one v -valuation function χ v such that the pair ( v, χ v ) is a vertex of some valuation V b ∈ Dom( p ). Denote by D the set of all such pairs (so D = (cid:83) Dom( p )). Thesame is true for the range and denote by R all pairs appearing as vertices invaluations in p (Dom( p )). It follows that p uniquely defines a g -compatible map q : D → R . Fix a big set S ∈ U . Then the set of pairs (cid:110)(cid:16) χ b ( S ) , χ (cid:48) g ( b ) ( g ( S )) (cid:17) : ( b, χ b ) ∈ D, q ( b, χ b ) = ( g ( b ) , χ (cid:48) g ( b ) ) (cid:111) is the graph of a partial permutation of { , , , , . . . , | S | − } fixing 0 if definedon it. Extend it to a permutation θ pS of { , , , , . . . , | S | − } fixing 0.Now we define ˆ q : V → V by mapping ( b, χ ) ∈ V to ( g ( b ) , χ (cid:48) ) such that χ (cid:48) ( g ( S ))) = θ pS ( χ ( S )) and ˆ p : V v → V g ( v ) where V g ( v ) is created from V v bymapping every vertex ( b, χ ) ∈ V v to ˆ q ( b, χ ).22t is easy to verify that ˆ q is a well defined permutation of V which extends q and is g -compatible. Moreover it preserves the relation of genericity between el-ements of V . Therefore also ˆ p is a well-defined permutation of C which preservesgeneric sets, extends p and is g -compatible. Consequently ˆ p is an automorphismof C .By Lemma 4.2 the extension property of C for partial isomorphisms of φ ( A )follows easily. Let p be a partial isomorphism of φ ( A ). We extend it to ˆ p ∈ Aut( C ). First extend φ − ◦ p (which is an partial automorphism of A ) toautomorphism g ∈ Aut( B − ). Clearly p is g -compatible and because domainand range are generic, by Lemma 4.2 p extends to g -compatible ˆ p ∈ Aut( C ).This shows that C is indeed an extension of φ ( A ).For coherence, we use a similar argument to that in [35, 36]. Given a coherenttriple ( f , g , h ) of partial automorphsims of A we first extend this to a coherenttriple ( f, g, h ) of automorphisms of B − . We let ( f , g , h ) be the coherent tripleof partial automorphisms of φ ( A ) induced by ( f , g , h ). Using Lemma 4.2 weextend f to an f -compatible ˆ f ∈ Aut( C ). Similarly we obtain extensions ˆ g ,ˆ h of g, h . In order to ensure that the triple ( ˆ f , ˆ g, ˆ h ) is coherent, we only needto ensure that, in the proof of Lemma 4.2, the permutations θ pS can be chosencoherently. More precisely, we want to ensure that θ gS θ fS = θ hS . As in [36], if weextend any partial permutation α on { , . . . , s } to a permutation by mapping { , . . . , s } \ Dom( α ) to { , . . . , s } \ α (Dom( α )) in an order-preserving way, thenwe obtain the required coherence.Finally we verify that C is faithful for irreducible substructures. Let D bean irreducible substructure of C . We first show that D is generic. Suppose notand that V a , V b ∈ D form a non-generic pair of vertices. Let E a = { V v ∈ D : V a (cid:54)∈ Cl D ( V v ) } . As closures are unary, this is a (proper) substructure of D .Similarly define E b . Note that E a ∪ E b = D : otherwise, there is V v ∈ D with V a , V b ∈ Cl D ( V v ) and then V a , V b ⊆ V v , so form a generic pair. Moreover,no relation of C can involve a vertex V u ∈ E a \ E b and a vertex V v ∈ E b \ E a as V b ⊆ V u and V a ⊆ V v , which implies that V u , V v is not a generic pair. Thus D is a free amalgam of the substructures E a and E b , which is a contradictionto its irreducibility. So D is generic.Because D is generic it follows that S = π ( D ) is small. Indeed, for each u ∈ S , there is a u -valuation χ u such that the set of pairs { ( u, χ u ) : u ∈ S } is generic. If S were big, this would imply that { χ u ( S ) : u ∈ S } has size | S | ,which is impossible (its elements j satisfy 1 ≤ j < | S | ).It follows that there is g ∈ Aut( B − ) such that g ( π ( D )) ⊆ A . The map p : D → φ ( A ) given by p ( V u ) = φ ( g ( u )) is a g -compatible partial automorphismof C with generic domain and range. By Lemma 4.2, p extends to ˆ p ∈ Aut( C )and ˆ p ( D ) ⊆ φ ( A ). This completes the proof that C is faithful for irreduciblesubstructures. 23 emark. The construction above adds an extra tool to the existing construc-tions of EPPA-extensions and can be thus used as an additional layer in theconstruction of EPPA-extensions for non-free amalgamation classes based onapplication of Herwig-Lascar theorem [14, 38, 33]. An example of such applica-tion is given in [3] giving EPPA for some classes of antipodal metric spaces.
In this section we discuss how some previously-studied classes of structurescan be viewed naturally as free amalgamation classes of structures with partialfunctions.Before doing this, we mention an alternative viewpoint for classes of struc-tures with closures which we used in [11]. In the following examples we willshow how this is related to our definitions.Consider a class K of finite L -structures, closed under isomorphisms, anda distinguished class (cid:118) of embeddings between elements of K , called strongembeddings . We shall assume (cid:118) is closed under composition and contains allisomorphisms. In this case, we refer to ( K ; (cid:118) ) as a strong class . If A is asubstructure of B ∈ K and the inclusion map A → B is in (cid:118) , then we say that A is a strong substructure of B and write A (cid:118) B . In the other words, a strongclass is a subcategory of K with the strong embeddings.The Ramsey property and amalgamation property can be defined analo-gously to the Ramsey property and amalgamation property of classes of L -structures, but considering only strong substructures and strong embedding.Most of the Fra¨ıss´e theory remains unaffected in this setting (see [11] for moredetails). k -orientations For a fixed natural number k , a k -orientation is an oriented (that is, directed)graph such that the out-degree of every vertex is at most k . We say that asubstructure G = ( V , E ) of a k -orientation G = ( V , E ) is successor closed if there is no edge from V to V \ V in G .Denote by D k the class of all finite k -orientations. This is a hereditaryclass closed for free amalgamation over successor-closed subgraphs and thusthe successor-closedness plays the rˆole of strong substructure, so D k can beconsidered as a class with corresponding strong embeddings. We show how toturn D k into a free amalgamation class in the sense of Definition 1.1.Given an oriented graph G = ( V, E ) ∈ D k denote by G + the structure withvertex set V and (partial) unary functions F , F , . . . , F k . The function F i ,1 ≤ i ≤ k , is defined for every vertex of out-degree i and maps the vertex to allvertices in its out-neighborhood. Denote by D + k the class of all structures G + for G ∈ D k . Because G +1 is a substructure of G +2 if and only if G is successor24losed in G it follows that D + k is a free amalgamation class. We immediatelyobtain: Theorem 5.1.
The class −→D + k is Ramsey and there exists a class O k ⊆ −→D + k with the ordering property (with respect to D + k ). The class D + k has the extensionproperty for partial automorphisms. Let us briefly discuss what is the structure of O k . Given −→ A ∈ O k , the closure-components (recall Definition 3.1) of −→ A corresponds to strongly connected com-ponents in the underlying oriented graph and −→ A is an ordered closure-extensionof level 0 if and only if the underlying graph is strongly connected. More gener-ally −→ A is an ordered closure-extension of level k if it contains a single stronglyconnected component C of level k and all other vertices of −→ A are reachable from C via an oriented path. Condition A6 of Definition 3.2 thus requires that the or-dering of C is determined by the isomorphism type of A (the underlying orientedgraph) and the ordering of A \ C . Thus in O k , vertices are ordered primarilyby the number of vertices in their closure. Every closure-component forms aninterval where the order within this interval is fixed by the similarity type ofcorresponding closure-extension. The relative order of closure-components isgiven by their isomorphism type and the ordering of closure-components reach-able from them. This can be seen as a generalization of the order of orientedforests described in Section 3.Theorem 5.1 can be seen as the most elementary use of Theorems 1.3, 1.4,and 1.7, but it has important consequences. Denote by C k the undirected reductsof oriented graphs in D k , that is, the class of all unoriented graphs which canbe oriented to an k -orientation. Given a graph G = ( V, E ), its predimension is δ ( G ) = k | V | − | E | . It is the heart of Hrushovski predimension construction thatthe class C k forms a free amalgamation class for the following notion of strongsubgraph. Given a graph G ∈ C k its subgraph H is a self sufficient or strongsubgraph if for every subgraph H (cid:48) of G containing H it holds that δ ( H ) ≤ δ ( H (cid:48) ).The connection between the Hrushovski predimension construction and ori-entability follows by the Marriage Theorem and was first introduced in [9, 10].Its consequences in Ramsey theory are the main topic of [11] and they are outof scope of this paper. We however point out why this free amalgamation classover strong subgraphs does not translate to a free amalgamation class when en-riched by partial functions representing the smallest self-sufficient subgraph of agiven set. Consider a graph in C created as amalgamation depicted in Figure 6.While in both B and B the vertices denoted by circles forms a self-sufficientsubstructures, it is not the case in the free amalgamation. The predimension ofthe 4 independent vertices is 8, while the predimension of the whole amalgamis 6. It follows that in order to represent self-sufficient substructures by meansof partial functions, a new function from the vertices denoted by circles wouldneed to be added. This makes the amalgamation non-free in our representationand this is the reason why additional information about orientation of the edgesis needed. 25 B A Figure 6: An amalgamation of B , B ∈ C over A .Figure 7: Fano plane (Steiner (2,3)-system). It was was established in [4] that the class of finite partial Steiner systems isRamsey with respect to strong subsystems. Moreover, the ordering propertyfollows from techniques of [31]. We derive both results by a re-interpretation ofpartial Steiner systems as a free amalgamation class in a functional language.This is an example where non-unary functions are necessary.For fixed integers r ≥ t ≥
2, by a partial Steiner ( r, t ) -system we mean an r -uniform hypergraph G = ( V, E ) with the property that ever t -element subsetof V is contained in at most one edge of G (if there is exactly one such edge, wehave a Steiner system). Abusing terminology somewhat, we shall refer to thissimply as a Steiner ( r, t )-system. Given two Steiner ( r, t )-systems G and H , wesay that G is a strongly induced subsytem of H if1. G is an induced subhypergraph of H ; and,2. every hyperedge of H which is not a hyperedge of G intersects G in atmost t − Definition 5.1.
Denote by S r,t the class of all finite structures A with onepartial function F from t -tuples to r -sets with the following properties:1. Every t -tuple (cid:126)x ∈ Dom( F A ) has no repeated vertices.2. For every (cid:126)x ∈ Dom( F A ) it holds that every vertex of (cid:126)x is in F A ( (cid:126)x ) andevery t -tuple (cid:126)x of distinct vertices of F A ( (cid:126)x ) is in Dom( F A ) and F A ( (cid:126)x ) = F A ( (cid:126)x ). 26igure 8: The bowtie graph.It is easy to see that S r,t is a free amalgamation class.Given a Steiner ( r, t )-system G = ( V, E ) we can interpret it as a structure S G ∈ S r,t with vertex set V and function F ( (cid:126)x ) defined for every t -tuple (cid:126)x ofdistinct vertices such that there is hyperedge A ∈ E containing all vertices of (cid:126)x .In this case we put F ( (cid:126)x ) = A .Observe that if G is a strong subsystem of H if and only if S G is a substruc-ture of S H . It follows that Steiner ( r, t )-systems are in 1–1 correspondence tostructures S G and moreover this correspondence maps subsystems to substruc-tures.We obtain an alternative proof of the following main result of [4]: Theorem 5.2.
The class −→S r,t is a Ramsey class with the ordering property.Proof.
The Ramsey property follows directly from Theorem 1.3. For the or-dering property, note that single vertices are closed in structures in S r,t , so allorderings are admissible, in the sense of Theorem 1.4.In [21] we obtained further corollaries to this approach. We remark that theextension property for partial automorphisms is, to our knowledge, presentlyopen for the class of partial Steiner systems. Of course, our result does notapply in this case, as the function introduced is not unary. A bowtie B is a graph consisting of two triangles with one vertex identified (seeFigure 8). A graph G is bowtie-free if there is no monomorphism from B to G .The existence of a universal graph in the class of all countable bowtie-free graphswas shown in [24]. The paper [7] gave a far reaching generalization by giving acondition for the existence of ω -categorical universal graphs for classes definedby forbidden monomorphisms (which we refer to as to as Cherlin-Shelah-Shiclasses ). This led to several new classes being identified [8], [6], [5]. Bowtie-free graphs represent a key example of a class that is not a free amalgamationclass by itself, but can be turned into one by means of unary functions. Thisanalysis was carried in [19] where we gave an explicit characterisation of theultrahomogeneous lift and the Ramseyness of this lift. The presentation can begreatly simplified by considering structures with partial functions and moreoverwe show the extension property for partial automorphisms (See also [35] forrelated results on ample generics). 27hile not all Cherlin-Shelah-Shi classes give rise to free amalgamation classes(see the more detailed analysis in [20]), what follows can be generalized to manyof the other block-path examples given by [6] and [5].We review the main observations about the structure of bowtie-free graphsfrom [19]. For completeness we include the (easy) proofs.
Definition 5.2 (Chimneys) . For n ≥
2, an n -chimney graph , Ch n , is a freeamalgamation of n triangles over one common edge. A chimney graph is anygraph Ch n for some n ≥ K (a clique on 4 vertices) will form the only com-ponents of bowtie-free graphs formed by triangles. The assumption n ≥ Ch is not an induced subgraph of K . Definition 5.3 (Good bowtie-free graphs) . A bowtie-free graph G = ( V, E ) is good if every vertex is contained either in a copy of chimney or a copy of thecomplete graph K .The structure of bowtie-free graphs is captured by means of the followingthree lemmas: Lemma 5.3 ([19]) . Every bowtie-free graph G is an induced subgraph of somegood bowtie-free graph G (cid:48) .Proof. The graph G can be extended in the following way:1. For every vertex v not contained in a triangle add a new copy of Ch andidentify the vertex v with one of the vertices of Ch .2. For every triangle v , v , v that is not part of a 2-chimney nor a K , add anew vertex v and the triangle v , v , v turning the original triangle into Ch .It is easy to see that step 1 . cannot introduce a new bowtie.Assume, to the contrary, that step 2 . introduced a new bowtie. Furtherassume that v is the unique vertex of degree 4 of this new bowtie and conse-quently there is another triangle on vertex v in G . Because G is bowtie-free,this triangle must share a common edge with the triangle v , v , v and thereforethe triangle v , v , v is already part of a K or a 2-chimney in the original graph G . A contradiction.Now we are ready to describe how to turn the class of bowtie-free graphs intoa free amalgamation class. Our language L will consist of one binary relation R and unary functions F , F and F with arities r ( F ) = 1, r ( F ) = 2, r ( F ) = 3.For every good bowtie-free graph G = ( V, E ) denote by G + the L -structurewith vertex set V and relations and functions defined as follows:28igure 9: All isomorphism types of closure-extensions in B .1. ( u, v ) ∈ R G if and only if { u, v } is edge of G .2. F G ( v ) = u if and only if { u, v } is contained in at least two triangles of achimney.3. F G ( v ) = { u , u } if and only if { v, u , u } is a triangle of a chimney and v is not contained in multiple triangles.4. F G ( v ) = { u , u , u } if and only if { v, u , u , u } forms a 4-clique in G .Denote by B the class of all A -partite substructures of structures G + where G is a good bowtie-free graph. Theorem 5.4. B is a free amalgamation class.Proof. Let A , B , B (cid:48) ∈ B . Assume that A is a substructure of both B and B (cid:48) .We show that the free amalgamation C of B and B over A is in B .There are good bowtie free graphs G and G such that B ⊆ G +1 and B (cid:48) ⊆ G +2 . We claim that that the free amalgamation H of G and G over A is a good bowtie-free graph and H + is the free amlgam of G +1 and G +2 over A .As C is a substructure of H + , it then follows that C ∈ B .Because A is a substructure of both G +1 and G +2 , the functions F and F ensure that the free amalgamation preserves the structure of chimneys: ifa vertex of a chimney in G is identified with a vertex of a chimney in G (because it is in A ) then also the bases (i.e. the edges in multiple triangles)of these chimneys are contained in A , so are identified in H and the result isagain a chimney. Similarly F makes sure that a 4-clique containing a vertex of A is in A . Finally free amalgamation cannot introduce any new triangles andthus the free amalgamation is a good bowtie-free graph H and H + is the freeamalgam of G +1 and G +2 over A . Corollary 5.5.
The class B has the irreducible-structure faithful extensionproperty for partial automorphisms; −→B is a Ramsey class and there is B (cid:48) ⊆ −→B with the ordering property (with respect to B ). The class B (cid:48) can be easily derived from the Definition 3.2. There are onlythree types of closure-extensions in B depicted in Figure 9 (with arrows repre-senting functions F , F and F and circles denoting the vertices of maximallevel). It follows that vertices are ordered by size of their closures (here we makeuse of the definition of (cid:22) refining the order given by number of vertices). That29s vertices in bases of chimney are first, vertices in the top of chimneys next andvertices in 4-cliques last. Every vertex-closure forms an interval. Pair of ver-tices of level 1 (in the the top of chimneys) form homologous extensions if andonly if they belong to the same chimney. It follows that for every chimney theset of its top vertices forms an interval and the relative order of these intervalscorresponds to the relative order of corresponding bases. Remark.
The Ramsey property and an explicit description of the admissibleordering was given in [19]. The relational language used is however more com-plicated and does not preserve all automorphisms of the Fra¨ıss´e limit of B . Thismakes it unsuitable for the extension property for partial automorphisms. Theformulation here is a more optimized version.The argument above together with the observation that in the Fra¨ıss´e limit B of B we have that for every finite S ⊆ B , | Cl B ( S ) | ≤ | S | , also gives acompact proof for the existence of an ω -categorical countable universal bowtie-free graph. This bound follows from the fact that function F , F , F cannotcascade. The ω -categoricity follows from the fact that the orbit of S in Aut( B ) isfully determined by the isomorphism type of Cl B ( S ) and there are only finitelymany closures for every finite S . This, of course, is just a re-formulation of theargument in [7]. It would be interesting to extend Theorem 1.7 to a class of structureswhich include non-unary functions. Perhaps this is too much to ask as EPPAis presently open even in the case of Steiner triple systems (as we remark inSection 5.2). However note that our structures involve partial functions andthus the EPPA may be easier to prove. But even for partial triple systems theEPPA seems to be presently open. On the structural Ramsey theory side open problems include Ramsey prop-erties of finite lattices and other algebraic structures where the axioms (such asassociativity) are difficult to control in an amalgamation procedure. See [20, 3]for results on Ramsey classes. The paper [20] gives a recursive construction for Ramsey classes. Is theresimilar result for EPPA?
Acknowledgements:
We would like to thank to Daoud Siniora for several usefuldiscussions concerning clique faithful EPPA and the notion of coherency.30 eferences [1] Fred G. Abramson and Leo A. Harrington. Models without indiscernibles.
Journal of Symbolic Logic , 43:572–600, 1978.[2] Omer Angel, Alexander S Kechris, and Russell Lyons. Random orderingsand unique ergodicity of automorphism groups.
Journal of the EuropeanMath. Society , 16:2059–2095, 2014.[3] Andres Aranda, David Bradley-Williams, Jan Hubiˇcka, Miltiadis Karaman-lis, Michael Kompatscher, Matˇej Koneˇcn´y, and Micheal Pawliuk. Ramseyexpansions of metrically homogeneous graphs. arXiv:1707.02612 , page 56p,2017.[4] Vindya Bhat, Jaroslav Neˇsetˇril, Christian Reiher, and Vojtˇech R¨odl. ARamsey class for Steiner systems. arXiv:1607.02792, 2016.[5] Gregory Cherlin and Saharon Shelah. Universal graphs with one forbiddensubgraph: the generic case. In preparation.[6] Gregory Cherlin and Saharon Shelah. Universal graphs with a forbiddensubgraph: block path solidity.
Combinatorica , pages 1–16, 2013.[7] Gregory Cherlin, Saharon Shelah, and Niandong Shi. Universal graphs withforbidden subgraphs and algebraic closure.
Advances in Applied Mathemat-ics , 22(4):454–491, 1999.[8] Gregory Cherlin and Lasse Tallgren. Universal graphs with a forbiddennear-path or 2-bouquet.
Journal of Graph Theory , 56(1):41–63, 2007.[9] David M. Evans. Ample dividing.
The Journal of Symbolic Logic ,68(04):1385–1402, 2003.[10] David M. Evans. Trivial stable structures with non-trivial reducts.
Journalof the London Mathematical Society (2) , 72(2):351–363, 2005.[11] David M. Evans, Jan Hubiˇcka, and Jaroslav Neˇsetˇril. Automorphism groupsand Ramsey properties of sparse graphs. In preparation, 56 pages, 2017.[12] Alfred W. Hales and Robert I. Jewett. Regularity and positional games.
Transactions of the American Mathematical Society , 106:222–229, 1963.[13] Bernhard Herwig. Extending partial isomorphisms on finite structures.
Combinatorica , 15(3):365–371, 1995.[14] Bernhard Herwig and Daniel Lascar. Extending partial automorphismsand the profinite topology on free groups.
Transactions of the AmericanMathematical Society , 352(5):1985–2021, 2000.3115] Wilfrid Hodges.
Model theory
Bulletin of Symbolic Logic , 9(03):387–405, 2003.[18] Ehud Hrushovski. Extending partial isomorphisms of graphs.
Combinator-ica , 12(4):411–416, 1992.[19] Jan Hubiˇcka and Jaroslav Neˇsetˇril. Bowtie-free graphs have a Ramsey lift.to appear in Advances in Applied Mathematics, arXiv:1402.2700, 32 pages,2014.[20] Jan Hubiˇcka and Jaroslav Neˇsetˇril. All those Ramsey classes (Ram-sey classes with closures and forbidden homomorphisms). Submitted,arXiv:1606.07979, 58 pages, 2016.[21] Jan Hubiˇcka and Jaroslav Neˇsetˇril. Ramsey classes with closure operations(selected combinatorial applications). Accepted, arXiv: arXiv:1705.01924,16 pages, 2017.[22] Alexander S. Kechris, Vladimir G. Pestov, and Stevo Todorˇcevi´c. Fra¨ıss´elimits, Ramsey theory, and topological dynamics of automorphism groups.
Geometric and Functional Analysis , 15(1):106–189, 2005.[23] Alexander S. Kechris and Christian Rosendal. Turbulence, amalgamation,and generic automorphisms of homogeneous structures.
Proceedings of theLondon Mathematical Society , 94(2):302–350, 2007.[24] P´eter Komj´ath. Some remarks on universal graphs.
Discrete Mathematics ,199(1):259–265, 1999.[25] Jaroslav Neˇsetril. Ramsey classes and homogeneous structures.
Combina-torics, probability and computing , 14(1-2):171–189, 2005.[26] Jaroslav Neˇsetˇril. Metric spaces are Ramsey.
European Journal of Combi-natorics , 28(1):457–468, 2007.[27] Jaroslav Neˇsetˇril and Vojtˇech R¨odl. A structural generalization of the Ram-sey theorem.
Bulletin of the American Mathematical Society , 83(1):127–128, 1977.[28] Jaroslav Neˇsetˇril and Vojtˇech R¨odl. On a probabilistic graph-theoreticalmethod.
Proceedings of the American Mathematical Society , 72(2):417–421,1978. 3229] Jaroslav Neˇsetˇril and Vojtˇech R¨odl. Simple proof of the existence of re-stricted Ramsey graphs by means of a partite construction.
Combinatorica ,1(2):199–202, 1981.[30] Jaroslav Neˇsetˇril and Vojtˇech R¨odl. The partite construction and Ramseyset systems.
Discrete Mathematics , 75(1):327–334, 1989.[31] Jaroslav Neˇsetˇril and Vojtˇech R¨odl. Partitions of subgraphs. In MiroslavFiedler, editor,
Recent advances in graph theory , pages 413–423. Prague,1975.[32] Jaroslav Neˇsetˇril and Vojtˇech R¨odl. Partitions of finite relational and setsystems.
Journal Combinatorial Theory, Series A , 22(3):289–312, 1977.[33] Martin Otto. Finite groupoids, finite coverings and symmetries in finitestructures. arXiv:1404.4599, 2014.[34] Micheal Pawliuk and Miodrag Soki´c. Amenability and unique ergodicity ofautomorphism groups of countable homogeneous directed graphs. acceptedto Ergodic Theory and Dynamical Systems, 2016.[35] Daoud Siniora.
Automorphism Groups of Homogeneous Structures . PhDthesis, University of Leeds, March 2017.[36] Daoud Siniora and S(cid:32)lawomir Solecki. Coherent extension of par-tial automorphisms, free amalgamation, and automorphism groups. arXiv:1705.01888 , 2017.[37] S(cid:32)lawomir Solecki. Extending partial isometries.
Israel Journal of Mathe-matics , 150(1):315–331, 2005.[38] S(cid:32)lawomir Solecki. Notes on a strengthening of the Herwig–Lascar extensiontheorem. Unpublished note, 2009.[39] Anatoly M. Vershik. Globalization of the partial isometries of metric spacesand local approximation of the group of isometries of Urysohn space.