1 Introduction
Let be a (not necessarily finite) graph and be subsets of . We say that a function is a partial automorphism of if is isomorphism of and , the graphs induced by on and respectively. This notion naturally extends to arbitrary structures (see Section 2).
In 1992 Hrushovski [18] proved that for every finite graph there is a finite graph such that is an induced subgraph of and every partial automorphism of extends to an automorphism of . This property is, in general, called the extension property for partial automorphisms:
Definition 1.1.
Let be a class of finite structures. We say that has the extension property for partial automorphisms (or EPPA), also called the Hrushovski property, if for every there is such that an for every isomorphism of substructures of there is an automorphism of such that . We call such an EPPAwitness for .
Hrushovski’s proof was grouptheoretical, Herwig and Lascar [15] later gave a simple combinatorial proof by embedding into a Kneser graph. After this, the quest of identifying new classes of structures with EPPA continued with a series of papers including [13, 14, 15, 17, 32, 33, 8, 29, 4, 20, 25, 21, 9, 22].
Let be a graph. We say that a (partial) map is distancepreserving if whenever are in the domain of then the distance between and is the same as the distance between and . Clearly, every automorphism is distancepreserving. In 2005 Solecki [32] (and independently also Vershik [33]) proved that the class of graphs has a variant of EPPA for distancepreserving maps. Namely, they proved that for every finite graph there is a finite graph satisfying the following:

and furthermore whenever are vertices of , then the distance between and is the same in and in , and

every partial distancepreserving map of extends to an automorphism of .
It is not very convenient to talk about distancepreserving maps, because they are relative to a graph and this does not work well together with taking subgraphs. Given a graph , it is more natural to consider the metric space where is the pathdistance between and in (we will call this the pathmetric space of ). And this is in fact what Solecki and Vershik did — they proved EPPA for all (integervalued) metric spaces, which is equivalent to EPPA for graphs with distancepreserving maps.
Vershik’s proof is unpublished, Solecki’s proof uses a complicated general theorem of Herwig and Lascar [15, Theorem 3.2] about EPPA for structures with forbidden homomorphisms, Hubička, Nešetřil and the author [20] recently gave a simple selfcontained proof of Solecki’s result. There is also a group theoretical proof by Sabok [30] using a construction à la Mackey [27].
This paper continues in this direction. We prove the following theorem (antipodal metric spaces are finite metric subspaces of the pathmetric of countable antipodal metrically homogeneous graphs, see Section 2.3).
Theorem 1.2.
Every class of antipodal metric spaces from Cherlin’s list has EPPA.
2 Preliminaries
A (not necessarily finite) structure is homogeneous if every partial automorphism of with finite domain extends to a full automorphism of itself (so it is, in a sense, an EPPAwitness for itself). Gardiner proved [11] that the finite homogeneous graphs are disjoint unions of cliques of the same size, their complements, the 5cycle and which is the graph on the vertex set where is connected with if and only if and . Lachlan and Woodrow later [26]classified the countably infinite homogeneous graphs. These are disjoint unions of cliques of the same size (possibly infinite), their complements, the Rado graph, the free variants of the Rado graph and their complements.
Every homogeneous structure can be associated with the class of all (isomorphism types of) its finite substructures called its age. By the Fraïssé theorem [10], one can reconstruct the homogeneous structure back from this class (because it has the socalled amalgamation property). For more on homogeneous structures see the survey by Macpherson [28].
A graph is vertex transitive if for every pair of vertices there is an automorphism sending to , it is edge transitive if every edge can be sent to every other edge by an automorphism and it is distance transitive if for every two pairs of vertices in the same distance there is an automorphism sending the first onto the other.
Distance transitivity is a very strong condition. For example, there are only finitely many finite 3regular distance transitive graphs [5] and the full catalogue is available in some other particular cases. However, for larger degrees, the classification is unknown, see e.g. the book by Godsil and Royle [12] devoted to the study of distance transitive graphs.
In this paper we study metric homogeneity which is a combination of the notions of distance transitivity and homogeneity. A connected graph is metrically homogeneous if it is homogeneous as a metric space where the distances are the path distances. Cherlin [6] gave a list of countable metrically homogeneous graphs which is conjectured to be complete (and provably complete in some cases [1, 7]) in terms of classes of finite metric spaces which embed to the pathmetric space of the given metrically homogeneous graphs. All connected homogeneous graphs are also metrically homogeneous, because every pair of vertices is either connected by an edge or by a path of length 2. An example of nonhomogeneous metrically homogeneous graphs are cycles.
EPPA and other combinatorial properties of classes from Cherlin’s list were studied by Aranda, BradleyWilliams, Hng, Hubička, Karamanlis, Kompatscher, Pawliuk and the author [2, 3, 4] (see also [24]) and in [4]
almost all the questions were settled, only EPPA for antipodal classes of odd diameter and bipartite antipodal classes of even diameter remained open. An important step in this direction was later done by Evans, Hubička, Nešetřil and the author
[9] who proved EPPA for antipodal metric spaces of diameter 3 and upon whose ideas we are building here.Remark 2.1.
If one checks the known classes with EPPA, they will find out that they all are ages of homogeneous structures. This is not a coincidence. It is easy to see that if a class of finite structures has EPPA and the joint embedding property (for every there is which contains a copy of both of them) then is the age of a homogeneous structure. This restricts the candidate classes for EPPA severely and connects finite combinatorics with the study of infinite homogeneous structures and infinite permutation groups.
In the other direction, EPPA has some implications for the automorphism group (with the pointwise convergence topology) of the corresponding homogeneous structure, see for example the paper of Hodges, Hodkinson, Lascar, and Shelah [16].
2.1 structures
An important feature of the strengthening of the Herwig–Lascar theorem by Hubička, Nešetřil and the author is that it allows to also permute the language. However, it means that one needs to update all the standard definitions to accommodate this change.
The following notions are taken from [22], sometimes stated in a more special form which is sufficient for our purposes. Many of them were introduced by Hubička and Nešetřil in [23] (i.e. homomorphismembedding or completions).
Let be a language with relational symbols , each having associated arities denoted by and function symbols . All functions in this paper are unary and have unary range. Let be a permutation group on which preserves types and arities of all symbols. We will say that is a language equipped with a permutation group.
A structure is a structure with vertex set , functions for every and relations for every . We will write structures in bold and their corresponding vertex sets in normal font. Note that when is trivial, one obtains the usual notion of modeltheoretic structures and all the following notions also collapse to their standard variants.
The language and its permutation group is usually fixed and understood from the context (and it is in most cases denoted by and respectively). If the set is finite we call a finite structure. If the language contains no function symbols, we call a relational language and say that a structure is a relational structure.
A homomorphism is a pair where and is a mapping such that for every and we have:

, and,

.
For brevity we will also write for in the context where and for where . For a subset we denote by the set and by the homomorphic image of a structure . Note that we write to emphasize that respects the structure.
If is injective then is called a monomorphism. A monomorphism is an embedding if for every we have the equivalence in the definition, that is,
If is an embedding where is onto then is an isomorphism. If is an inclusion and is the identity then is a substructure of . For an embedding with being the identity we say that is a copy of in .
Given a structure and , the closure of in , denoted by , is the smallest substructure of containing . For we will also write for .
Generalising the notion of a graph clique, we say that a structure is irreducible if every pair of vertices of is together in some relation of (possibly with some more vertices). Note that this is a weaker notion than given in [22], but sufficient for the purposes of this paper (and giving the definition in full strength would require some more preliminary definitions).
Example.
In this paper the languages will only consist of unary and binary relations and unary functions. In this case a structure is irreducible if and only if the union of the binary relations is a clique.
Homomorphism is a homomorphismembedding if the restriction is an embedding whenever is an irreducible substructure of .
2.2 EPPA for structures
We now state the main result of [22], but first we need to give a few more definitions. Again, they are often EPPA variants of an analogous result for the Ramsey property (the Hubička–Nešetřil theorem [23]).
A partial automorphism of structure is an isomorphism where and are substructures of (remember that it also includes a full permutation of the language). We say that a class of finite structures has the extension property for partial automorphisms (EPPA) if for every there is such that is a substructure of and every partial automorphism of extends to an automorphism of . We call with such a property an EPPAwitness of . is an irreducible structure faithful EPPAwitness of if it has the property that for every irreducible substructure of there exists an automorphism of such that .
Definition 2.2.
Let be a structure. An irreducible structure is a completion of if there exists a homomorphismembedding such that it does not permute the language. It is automorphismpreserving if .
All structures in this paper will only consist of unary and binary relations (and possibly unary functions, but they will follow the existing binary relations). In this situation is a completion of if and (the structure induced by on ) differs from precisely by adding some binary relations on every pair of vertices of which is in none.
Definition 2.3.
Let be a finite language with relations and unary functions equipped with a permutation group . Let be a class of finite irreducible structures and a subclass of . We say that the class is a locally finite subclass of if for every and every there is a finite integer such that every structure has a completion provided that it satisfies the following:

For every vertex it holds that lies in a copy of ,

there is a homomorphismembedding from to , and,

every substructure of with at most vertices has a completion in .
We say that is a locally finite automorphismpreserving subclass of if in the condition above the completion of can always be chosen to be automorphismpreserving.
The main theorem of [22] is then as follows.
Theorem 2.4 ([22]).
Let be a finite language with relations and unary functions equipped with a permutation group , let be a class of irreducible finite structures which has EPPA and let be a hereditary locally finite automorphismpreserving subclass of with strong amalgamation. Then has EPPA. Moreover if EPPAwitnesses in can be chosen to be irreducible structure faithful then EPPAwitnesses in are irreducible structure faithful, too.
Here is hereditary if whenever and , then also . We will not define what strong amalgamation is (see [22]), but all classes for which we will use Theorem 2.4 will indeed satisfy this property.
We will also need the following theorem from [22].
Theorem 2.5 ([22]).
Let be a finite language with relations and unary functions equipped with a permutation group . Then the class of all finite structures has irreducible structure faithful EPPA.
2.3 Metrically homogeneous graphs
Most of the details of Cherlin’s metric spaces are not important for this paper. We only give minimum necessary definitions and facts and refer the reader to [6], [4] or [24].
All the metric spaces we will work with will have distances from for some integer . Therefore, we will view them interchangeably as pairs where is the metric, as relational structures with trivial and binary relations (distance is not represented) and as complete graphs with edges labelled by (we will call these complete edgelabelled graphs). The last point of view works well with the notion of completion: Given a (not necessary complete) edgelabelled graph , a complete edgelabelled graph is its completion if is a noninduced subgraph of .
We will also say that two vertices are in distance and that they are connected by an edge of length interchangeably. In particular, when we talk about an edge of a edgelabelled, graph, it can be an arbitrary pair of vertices such that their distance is defined, it does not necessarily mean that they are in distance 1.
Major part of Cherlin’s list of the classes of finite metric spaces which embed to the pathmetric of a countably infinite metrically homogeneous graph consists of certain 5parameter classes . These are classes of metric spaces with distances (we call the diameter of such spaces) such that some other triangles are forbidden (e.g. triangles of short odd perimeter or triangles of long even perimeter).
A special case of these classes are antipodal
classes, where the five parameters have only two degrees of freedom. Here we will denote the antipodal classes as
.^{1}^{1}1The other parameters are then defined as , , and . is defined as the class of all finite metric spaces with distances from such that they contain no triangle with distances such that one of the following holds:
,

is odd and , or

is odd and .
However, here we only need the following fact:
Fact 2.6 (Antipodal spaces).
In a class of antipodal metric spaces from Cherlin’s list the following holds:

The edges of length form a matching (that is, for every vertex there is at most one vertex in distance from it) and one can without loss of generality assume that it is a perfect matching.

For every pair of vertices such that and for every vertex we have .

If one selects exactly one vertex from each edge of length , the metric space they induce is from a special (nonantipodal) class of diameter which we will call .^{2}^{2}2It is in fact for , , and . And the other way around, one can get an antipodal metric space from every metric space by taking two disjoint copies of , connecting every vertex to its copy by an edge of length and using point 2 to fillin the missing distances.
There are two kinds of antipodal classes with different combinatorial behaviour — those that come from a countable bipartite metrically homogeneous graph and those that come from a nonbipartite one. We will call the first the bipartite classes (their members have the property that they contain no triangles, or more generally cycles, of odd perimeter) and we will call the others the nonbipartite ones. This is slightly misleading, because some of the finite metric subspaces of the pathmetric of a nonbipartite metrically homogeneous graph are surely bipartite, but it should not cause any confusion in this paper. The nonbipartite class of antipodal metric spaces of diameter is closely connected to switching classes of graphs and twographs (see [9]).
The following fact summarizes results from [4] about the nonbipartite odd diameter antipodal classes.
Fact 2.7.
Let be a nonbipartite class of antipodal metric spaces of odd diameter . Let be a edgelabelled graph such that edges of length of form a perfect matching and furthermore for every such that and either is not connected by an edge to any of , or . Suppose furthermore that contains none of the finitely many cycles forbidden in .
Let be a mapping satisfying the following.

Whenever is an edge of , then .

Let and be two different edges of length of . Then , and .
There is such that the following holds.

is a completion of with the same vertex set,

for every edge of it holds that , and

Every automorphism of which preserves values of is also an automorphism of .
Such can be constructed by picking one vertex from each edge of length , considering this auxiliary metric space of diameter , completing it using [4] and then pulling this completion back using . The proof then uses [4, Lemma 4.18] (see also [19]) and the observation that the completion procedure for from [4] preserves the equivalence “”. That is, we say that two edgelabelled graphs and are equivalent if they share the same vertex set and the same edge set and every edge has either the same label in both and , or it has label in and in . The completion procedure then produces equivalent graphs whenever given equivalent graphs.
3 The odd diameter nonbipartite case
EPPA for the even diameter nonbipartite case was proved in [4] (because it has automorphismpreserving completion by itself). In this section we prove the following proposition
Proposition 3.1.
Let be a nonbipartite class of antipodal metric spaces of odd diameter. Then for every there is which is an EPPAwitness of .
Fix . We can without loss of generality assume that every vertex has some vertex such that . Enumerate the edges of of length as and let be their indices, that is, (we will sometimes treat also as the set itself using the natural bijection). We furthermore denote , where and are vertices of .
3.1 The expanded language
We will call a function is a valuation function. For a set , we denote by the flip of , that is, the function defined as
and for a permutation of we denote by the function satisfying .
Let be the language consisting of symmetric binary relations representing the distances, a unary function , and unary marks for every and for every valuation function .
We now define the permutation group . Let be a permutation of and let be such that if , then also . For every we let be the set of such that . We denote by the permutation of sending and fixing and pointwise. is then the group consisting of all for every possible choice of and .
For notational convenience, whenever is a structure and is a vertex which has precisely one unary mark , we will denote by its projection and by its valuation. If does not have precisely one unary mark, we leave and undefined.
The following observation follows directly from the definitions above.
Observation 3.2.
Let be a structure such that every vertex of has precisely one unary mark , let be an automorphism of , let be arbitrary vertices of , and assume that and . Then we have
if and only if
This implies that the function defined by if and otherwise is invariant under automorphism of (including permuting the language).
3.2 The class and completion to it
Let . We say that a structure is a suitable expansion of if the following hold:

and share the same vertex set,

and share the same relations ,

if and only if ,

every vertex of has precisely one unary mark,

if and in , then , where , and

in it holds that if and only if is odd.
Denote by the class of all suitable expansions of all where the edges of length form a perfect matching (Fact 2.6 says that this is without loss of generality; one can always uniquely and canonically add vertices so that this condition is satisfied).
Proposition 3.3.
is a locally finite automorphismpreserving subclass of , the class of all finite structures.
Proof.
Let be a large enough integer (say, at least 4 and at least twice the number of vertices of the largest forbidden cycle in ) and let and be as in Definition 2.3. (This is a different than we fixed at the beginning of this section.) We will not need here.
The fact that for every it holds that lies in a copy of implies that if and only if and furthermore the edges of length form a perfect matching in (because this holds in ).
Every substructure on at most vertices having a completion in implies the following:

Every pair of vertices is in at most one distance relation (and these relations are symmetric),

every vertex of is in precisely one unary relation,

whenever and , then .
We can assume that if and is a vertex of such that at least one of , is defined, then in fact both distances are defined and furthermore , because there is a unique way how to complete it. It also follows that whenever are vertices such that their distance is defined, then if and only if is odd.
Finally, from the definition of it also follows that contains no cycles forbidden in (we needed to be twice the number of vertices because Definition 2.3 talks about substructures and these need to be closed for functions). Hence if we define function as if and otherwise, Fact 2.7 gives us an automorphismpreserving way to add the remaining non distances, which is exactly what we need for a completion to . ∎
Let us remark that is hereditary and from Fact 2.7 it follows that it is a strong amalgamation class.
3.3 Constructing the witness
For we define by putting
We define a structure which is a expansion of . It will have the same vertex set and the same distance relations as , but we furthermore put for every such that . For every we put and .
3.4 Extending partial automorphisms
We will show that extends all partial automorphisms of . Fix a partial automorphism of . Without loss of generality we can assume that whenever and , then also (because there is a unique way of extending to ). Let be a permutation of extending the action of on the edges of length of in the natural sense.
We now define set of flipping pairs. We put and in if and in . Note that if both and are in the domain of then the outcome would be the same if we considered instead (because is an automorphism and therefore preserves the parity of and thus also the (non)equality of the corresponding valuations). Note also that if we considered instead of , the outcome would still be the same.
It is easy to check that the pair is a automorphism of and hence it extends to an automorphism of , where and . But this means that is an automorphism of extending and we thus conclude the proof of Proposition 3.1.
3.5 Remarks

The same strategy would also work for proving EPPA for antipodal metric spaces of even diameter, we would only need to pick a subset such that and precisely one of is in for every and replace each occurrence of “odd distance” by “distance from ” and “even distance” by “distance such that ”. (Note that for even , we have , so “is both odd and even” in this sense.)

Cherlin also allows to forbid certain sets of valued metric spaces (he calls them Henson constraints). We chose not to include these classes in order to avoid further technical complications, but using irreducible structure faithfulness and the fact that the completion from Fact 2.7 does not create distances and gives EPPA also in this case.
4 The even diameter bipartite case
The odd diameter bipartite case was done in [4] (because the edges of length go across the parts and there is thus an automorphismpreserving completion), so it suffices to deal with the even diameter case. We prove the following proposition.
Proposition 4.1.
Let be a bipartite class of antipodal metric spaces of even diameter. Then for every there is which is an EPPAwitness of .
The structure of the proof will be very similar to the odd nonbipartite case. We will also introduce some facts from [4] about completions, add unary functions and unary marks which will help us decide how to fillin the missing distances while preserving all necessary automorphisms. We have to be a bit more careful in dealing with the bipartiteness (edges of length now lie inside the parts, so we need to make preserve the bipartition, there are also infinitely many forbidden cycles — the odd perimeter ones), but the general structure is identical.
Fix . We can without loss of generality assume that every vertex has some vertex such that . Consider the set of edges of of length and let be their indices, that is, . We denote , where and are vertices of and also assume that , where consists of the indices of edges in one part and of the indices of edges in the other part.
We also assume without loss of generality that (otherwise we can add more vertices to , and if this larger structure has an EPPAwitness , then it is also an EPPAwitness of the original ).
We will need the following analogue of Fact 2.7.
Fact 4.2.
Let be a bipartite class of antipodal metric spaces. Let be a edgelabelled graph such that edges of length of form a perfect matching and furthermore for every such that and either is not connected by an edge to any of , or . Suppose furthermore that contains no odd perimeter cycles and none of the finitely many even perimeter cycles forbidden in .
Let be a set such that and exactly one of is in for every and denote by the set .
Let be a mapping satisfying the following.

Whenever is an edge of , then implies that and implies that .^{3}^{3}3This seemingly sloppy statement is necessary in order to deal with being in both and for even .

Let and be two different edges of length of . Then , and .
There is such that the following holds.

is a completion of with the same vertex set,

for every edge of it holds that implies that and implies that , and

Every automorphism of which preserves values of is also an automorphism of .
4.1 The expanded language
As in the odd nonbipartite case, we will call a function a valuation function, adopt the same notions of flips and permutations . We also let be the same language as before.
We also keep the definition of , however, we will put to be the group consisting of all for every possible choice of , but only for such permutations which preserve the partition of into two parts (that is, either and , or and vice versa.
Again, for a vertex in a structure which has precisely one unary mark , we define and and we have a very similar observation as before.
Observation 4.3.
Let be a structure (not necessarily a bipartite one) such that every vertex of has precisely one unary mark , let be an automorphism of , let be arbitrary vertices of , and assume that and . Then we have
if and only if
This implies that the function defined by if and otherwise is invariant under automorphism of (including permuting the language).
4.2 The class and completion to it
Now we also have to ensure that structures from are bipartite. Let . We say that a structure is a suitable expansion of if the following hold:

and share the same vertex set,

and share the same relations ,

if and only if ,

every vertex of has precisely one unary mark,

if and in , then ,

in it holds that if then and if then , and

let and (where is taken with respect to ). Then is the bipartition of (and thus also of ).
Denote by the class of all suitable expansions of all where the edges of length form a perfect matching.
Proposition 4.4.
is a locally finite automorphismpreserving subclass of , the class of all finite structures.
Proof.
Let be a large enough integer (say, at least 4 and at least twice the number of vertices of the largest evenperimeter forbidden cycle in ) and let and be as in Definition 2.3. (This is a different then fixed at the beginning of this section.) We will not need here.
Like for the odd nonbipartite case we get the following:

Every vertex of is in precisely one unary relation,

every pair of vertices is in at most one distance relation (and these relations are symmetric),

,

the edges of length form a perfect matching in ,

whenever and , then ,

if and is a vertex of such that at least one of , is defined, then both distances are defined and furthermore .

let be vertices such that their distance is defined. Then implies and implies .
Furthermore, from the last condition for a suitable expansion we also get that whenever two vertices of are in an even distance, then there is such that and that whenever they are in an odd distance, then and are in different parts of . Note that this implies that contains no cycles of odd perimeter (each cycle has to contain an even number of odd edges).
Finally, from the definition of it also follows that contains no even cycles forbidden in . Hence if we define function as if and otherwise, Fact 4.2 gives us an automorphismpreserving way to add the remaining distances, which is exactly what we need for a completion to . ∎
Let us again remark that is hereditary and from Fact 4.2 it follows that it is a strong amalgamation class.
4.3 Constructing the witness
This is completely the same as for the odd diameter nonbipartite case. We define a structure which is a expansion of with the same vertex set, use Theorems 2.4 and 2.5 with Proposition 4.4 to get which is an EPPAwitness for . Finally, we put to be the reduct of forgetting all unary marks and all functions .
4.4 Extending partial automorphisms
Again, this is completely the same as before with the exception that the permutation of has to preserve the bipartition (with, possibly, exchanging and ). Every partial automorphism of respects the bipartition, and since we assumed that , it is always possible to extend it to a full permutation as needed.
Let us remark that if one is a bit more careful, the same strategy again gives coherent EPPA.
5 Conclusion
We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2.
We think of this paper as the first example of a more general method for bypassing the lack of an automorphismpreserving completion. As observed in [4], the nonbipartite antipodal classes of odd diameter do not have automorphismpreserving completion essentially because there is no canonical way of completing the graph consisting only of two edges of length — one pair of the added distances has to be odd and the other even and there is no way of picking this canonically.
Here, we are using the method of valuation functions to add more information to the structures (and thus restrict automorphisms) while preserving all partial automorphisms of one given structure and then putting this expanded class into the existing machinery. A similar trick can be done also for structures with higher arities using higher arity valuation functions (cf. [22]). However, there are still classes where this method does not work, for example the class of tournaments which poses a longstanding important problem in this area.
6 Acknowledgements
I would like to thank Jan Hubička, Jaroslav Nešetřil and Gregory Cherlin for valuable advice and comments which significantly improved this paper. Supported by project 1813685Y of the Czech Science Foundation (GAČR).
References
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