EPPA for two-graphs and antipodal metric spaces

TL;DR

This paper proves that finite two-graphs and certain antipodal metric spaces have the extension property for partial automorphisms, providing new insights into their automorphism structures and answering longstanding questions.

Contribution

It establishes EPPA for finite two-graphs and antipodal metric spaces using a combinatorial proof, addressing open problems and highlighting unique behaviors of two-graphs.

Findings

EPPA holds for finite two-graphs

EPPA holds for integer valued antipodal metric spaces of diameter 3

Two-graphs exhibit unique properties differing from other classes with EPPA

Abstract

We prove that the class of finite two-graphs has the extension property for partial automorphisms (EPPA, or Hrushovski property), thereby answering a question of Macpherson. In other words, we show that the class of graphs has the extension property for switching automorphisms. We present a short, self-contained, purely combinatorial proof which also proves EPPA for the class of integer valued antipodal metric spaces of diameter 3, answering a question of Aranda et al. The class of two-graphs is an important new example which behaves differently from all the other known classes with EPPA: Two-graphs do not have the amalgamation property with automorphisms (APA), their Ramsey expansion has to add a graph, it is not known if they have coherent EPPA and even EPPA itself cannot be proved using the Herwig--Lascar theorem.

Figures

Equations

$$\mathrm{Dom}(f)=\mathrm{Dom}(h),\mathrm{Range}(f)=\mathrm{Dom}(g),\mathrm{Range}(g)=\mathrm{Range}(h)$$

$$h=g\circ f.$$

$${\chi }_i(e_j)=\{\begin{array}{ll}0& \text{if }j\ge i\\ 0& \text{if }j<i\text{ and }{d}_{\mathbf{A}}(x_i,x_j)=1\\ 1& \text{otherwise.}\end{array}$$

$$\theta ((e,\chi ))=(\hat{\phi }(e),\xi )$$

$$\xi (\hat{\phi }(f))=\{\begin{array}{ll}\chi (f)& \text{if }\{e,f\}\notin F\\ 1-\chi (f)& \text{if }\{e,f\}\in F.\end{array}$$

$$(\{e,f\}\notin F_1\text{mbox}and\{{\hat{\phi }}_1(e),{\hat{\phi }}_1(f)\}\notin F_2)\text{mbox}or(\{e,f\}\in F_1\text{mbox}and\{{\hat{\phi }}_1(e),{\hat{\phi }}_1(f)\}\in F_2).$$

$${\xi }_2({\hat{\phi }}_2({\hat{\phi }}_1(f)))=\{\begin{array}{ll}{\xi }_1({\hat{\phi }}_1(f))& \text{if }\{{\hat{\phi }}_1(e),{\hat{\phi }}_1(f)\}\notin F_2\\ 1-{\xi }_1({\hat{\phi }}_1(f))& \text{if }\{{\hat{\phi }}_1(e),{\hat{\phi }}_1(f)\}\in F_2.\end{array}$$

$$\overrightarrow{\mathbf{C}}⟶(\overrightarrow{\mathbf{B}}{)}_2^{\overrightarrow{\mathbf{E}}}$$

Full text

The class of two-graphs is an important new example which behaves differently from all the other known classes with EPPA: Two-graphs do not have the amalgamation property with automorphisms (APA), their Ramsey expansion has to add a graph, it is not known if they have coherent EPPA and even EPPA itself cannot be proved using the Herwig–Lascar theorem.

1. Introduction

Two-graphs, introduced by G. Higman and studied extensively since the 1970s [Cam99, Sei73], are 3-uniform hypergraphs with the property that on every four vertices there is an even number of hyperedges. A class $\mathcal{C}$ of finite structures (such as hypergraphs) has the extension property for partial automorphisms (EPPA, sometimes also called Hrushovski property) if for every $\mathbf{A}\in\mathcal{C}$ there exists $\mathbf{B}\in\mathcal{C}$ containing $\mathbf{A}$ as an (induced) substructure such that every isomorphism between substructures of $\mathbf{A}$ extends to an automorphism of $\mathbf{B}$. We call $\mathbf{B}$ an EPPA-witness for $\mathbf{A}$. We prove:

Theorem 1.1**.**

The class $\mathcal{T}$ of all finite two-graphs has EPPA.

Our result answers a question of Macpherson which is also stated in Siniora’s PhD thesis [Sin17] and can be seen as a contribution to the ongoing effort of identifying new classes of structures with EPPA. This was started in 1992 by Hrushovski’s proof [Hru92] that the class of all finite graphs has EPPA, and followed by a series of papers dealing with other classes, including [ABWH*+*17, Con19, Her95, Her98, HL00, HO03, HKN19, HKN18, Kon19a, Ott17, Sol05, Ver08].

All proofs of EPPA in this paper are purely combinatorial and self-contained. The second part of the paper requires some model-theoretical notions and discusses in more detail the interplay of the following properties for which there were no known examples before:

1. (1)

The usual procedure for building an EPPA-witness is to construct an incomplete object (where some relations are missing) and later complete it to satisfy axioms of the class without affecting any automorphisms (i.e. one needs to have an automorphism-preserving completion [ABWH*+*17]). This is not possible for two-graphs and thus makes them related to tournaments which pose a well known open problem in the area, see Remark 8.1. 2. (2)

The class $\mathcal{T}$ does not have APA (amalgamation property with automorphisms). Hodges, Hodkinson, Lascar, and Shelah [HHLS93] introduced this notion and showed that APA together with EPPA imply the existence of ample generics (see also [Sin17, Chapter 2]). To the authors’ best knowledge, $\mathcal{T}$ is the only known class with EPPA but not APA besides pathological examples, see Section 6. 3. (3)

In all cases known to the authors except for the class of all finite groups, whenever a class of structures $\mathcal{C}$ has EPPA then expanding a variant of $\mathcal{C}$ by linear orders gives a Ramsey expansion. This does not seem to be the case for two-graphs, see Section 7. 4. (4)

Solecki and Siniora [SS19, Sol09] introduced the notion of coherent EPPA (see Section 2.1) as a way to prove that the automorphism group of the respective Fraïssé limit contains a dense locally finite subgroup. Our method does not give coherent EPPA for $\mathcal{T}$ and thus it makes $\mathcal{T}$ the only known example with EPPA for which coherent EPPA is not known. However, our method does give coherent EPPA for the class of all antipodal metric spaces of diameter 3 and using it we are able to obtain a dense locally finite subgroup of the Fraïssé limit of $\mathcal{T}$, see Section 5.

Two-graphs are closely related to the switching classes of graphs and to double covers of complete graphs [Cam99, Sei73], which is in fact key in this paper. Our results can thus be interpreted as a direct strengthening of the theorem of Hrushovski [Hru92] which states that the class of all finite graphs $\mathcal{G}$ has EPPA. Namely we can consider $\mathcal{G}$ with a richer class of mappings — the switching automorphisms.

Given a graph $\mathbf{G}$ with vertex set $G$ and $S\subseteq G$, the (Seidel) switching $\mathbf{G}_{S}$ of $\mathbf{G}$ is the graph created from $\mathbf{G}$ by complementing the edges between $S$ and $G\setminus S$. (That is, for $s\in S$ and $t\in G\setminus S$ it holds that $\{s,t\}$ is an edge of $\mathbf{G}_{S}$ if and only if $\{s,t\}$ is not an edge of $\mathbf{G}$. Edges and non-edges with both endpoints in $S$ or $G\setminus S$ are preserved.)

Given a graph $\mathbf{H}$ with vertex set $H$, a function $f\colon G\to H$ is a switching isomorphism of $\mathbf{G}$ and $\mathbf{H}$ if there exists $S\subseteq G$ such that $f$ is an isomorphism of $\mathbf{G}_{S}$ and $\mathbf{H}$. If $\mathbf{G}=\mathbf{H}$ we call such a function a switching automorphism.

Definition 1.1**.**

We say that a class $\mathcal{C}\subseteq\mathcal{G}$ has the extension property for switching automorphisms if for every $\mathbf{G}\in\mathcal{C}$ there exists $\mathbf{H}\in\mathcal{C}$ containing $\mathbf{G}$ as an induced subgraph such that every switching isomorphism of induced subgraphs of $\mathbf{G}$ extends to a switching automorphism of $\mathbf{H}$ and, moreover, every isomorphism of induced subgraphs of $\mathbf{G}$ extends to an automorphism of $\mathbf{H}$.

In this language we prove:

Theorem 1.2**.**

The class of all finite graphs $\mathcal{G}$ has the extension property for switching automorphisms.

Because of the ‘moreover’ part of Definition 1.1, Theorem 1.2 implies the theorem of Hrushovski. It is also a strengthening of Theorem 1.1 by the following well-known correspondence between two-graphs and switching classes [Cam99, Sei73].

Given a graph $\mathbf{G}$, its associated two-graph $T(\mathbf{G})$ is a two-graph on the same vertex set as $\mathbf{G}$ such that $\{a,b,c\}$ is a hyperedge if and only if the three-vertex subgraph induced by $\mathbf{G}$ on $\{a,b,c\}$ has an odd number of edges. Then a function $f\colon G\to H$ between graphs $\mathbf{G}$ and $\mathbf{H}$ is a switching isomorphism if and only if it is an isomorphism between the associated two-graphs. Thus, the existence of a switching isomorphism is an equivalence relation on the class of graphs and two-graphs correspond to the equivalence classes (called switching classes of graphs).

We shall see that the most natural setting for our proof is to work with the class of all finite integer-valued antipodal metric spaces of diameter 3. Following [ACM16] we call a metric space an integer-valued metric space of diameter 3 if the distance of every two distinct points is 1, 2 or 3. It is antipodal if

1. (1)

it contains no triangle with distances $2,2,3$, and 2. (2)

the edges with label $3$ form a perfect matching (in other words, for every vertex there is precisely one antipodal vertex at distance 3).

We require that the domain and image of a partial automorphism of such a structure should be closed under taking antipodal points. However, this can be assumed without loss of generality because there always is a unique way of extending a partial automorphism not satisfying this condition to one which does. In this language, we can then state our main theorem as:

Theorem 1.3**.**

The class of all finite integer-valued antipodal metric spaces of diameter 3 has coherent EPPA.

This theorem also holds for all other antipodal metric spaces from Cherlin’s catalogue of metrically homogeneous graphs [Che17], see [Kon19b]. It answers affirmatively a question of Aranda, Bradley-Williams, Hubička, Karamanlis, Kompatscher, Konečný and Pawliuk [ABWH*+*17] and completes the analysis of EPPA for all classes from Cherlin’s catalogue. However, in this note, for brevity, we often refer to an antipodal, integer-valued metric space of diameter 3 as an antipodal metric space. Other antipodal metric spaces are not considered.

2. Notation and preliminaries

It is in the nature of this paper to consider multiple types of structures. We will use bold letter such as $\mathbf{A},\mathbf{B},\mathbf{C},\ldots$ to denote structures ((hyper)graphs or metric spaces defined below) and corresponding normal letters, such as $A,B,C,\ldots$, to denote corresponding vertex sets. Our substructures (sub-(hyper)graphs or subspaces) will always be induced.

Formally, we will consider a metric space to be a complete edge-labelled graph (that is, a complete graph where edges are labelled by the respective distances), or, equivalently, a relational structure with multiple binary relations representing the distances. This justifies that we will speak of pairs of vertices at distance $d$ as of edges of length $d$. We will, however, use both notions (a vertex set with a distance function or a complete edge-labelled graph) interchangeably. We adopt the standard notion of isomorphism, embedding and substructure.

2.1. Coherent EPPA

Suppose $\mathcal{C}$ is a class of finite structures which has the hereditary and joint embedding properties. If $\mathcal{C}$ has EPPA, then it has the amalgamation property, so, assuming that there are only countably many isomorphism types of structures in $\mathcal{C}$, then we can consider the Fraïssé limit $\mathbf{M}$ of $\mathcal{C}$. Coherence is a natural strengthening of EPPA which guarantees that $\mathop{\mathrm{Aut}}\nolimits(\mathbf{M})$ has a dense locally finite subgroup [Sol09, SS19]. Note that the existence of a dense locally finite subgroup of the automorphism group of a homogeneous structure implies that its age has EPPA, but it is not known whether coherent EPPA also follows from this: see Section 5.1 of [SS19]. At the moment all previously known EPPA classes are also coherent EPPA classes. Interestingly, we can prove coherent EPPA for the antipodal metric spaces of diameter 3, but not for two-graphs (this is discussed in Section 5). We need to introduce two additional definitions.

Definition 2.1** (Coherent maps [Sol09, SS19]).**

Let $X$ be a set and $\mathcal{P}$ be a family of partial bijections between subsets of $X$. A triple $(f,g,h)$ from $\mathcal{P}$ is called a coherent triple if

[TABLE]

and

[TABLE]

Let $X$ and $Y$ be sets, and $\mathcal{P}$ and $\mathcal{Q}$ be families of partial bijections between subsets of $X$ and between subsets of $Y$, respectively. A function $\varphi\colon\mathcal{P}\to\mathcal{Q}$ is said to be a coherent map if for each coherent triple $(f,g,h)$ from $\mathcal{P}$, its image $(\varphi(f),\varphi(g),\varphi(h))$ in $\mathcal{Q}$ is coherent. In the case of EPPA, where the elements of $\mathcal{Q}$ are automorphisms of $Y$, we sometimes refer to the image of $\varphi$ as a coherent family of automorphisms extending $\mathcal{P}$.

Definition 2.2** (Coherent EPPA [Sol09, SS19]).**

A class $\mathcal{C}$ of finite structures is said to have coherent EPPA if $\mathcal{C}$ has EPPA and moreover the extension of partial automorphisms is coherent. More precisely, for every $\mathbf{A}\in\mathcal{C}$, there exists an EPPA-witness $\mathbf{B}\in\mathcal{C}$ for $\mathbf{A}$ and a coherent map $f\mapsto\hat{f}$ from the family of partial automorphisms of $\mathbf{A}$ to the group of automorphisms of $\mathbf{B}$. In this case we also call $\mathbf{B}$ a coherent EPPA-witness of $\mathbf{A}$.

3. EPPA for antipodal metric spaces

Given an antipodal metric space $\mathbf{A}$ we give a direct construction of a coherent EPPA-witness $\mathbf{B}$. Some ideas are based on a construction of Hodkinson and Otto [HO03] and some of the terminology is loosely based on Hodkinson’s exposition of this construction [Hod02]. Note that our techniques also give a very simple and short proof of EPPA for graphs.

Fix a (finite) antipodal metric space $\mathbf{A}$. Denote by $M=\{e_{1},e_{2},\ldots,e_{n}\}$ the set of all edges of $\mathbf{A}$ of length 3. For a function $\chi\colon M\to\{0,1\}$ we denote by $1-\chi$ the function satisfying $(1-\chi)(e)=1-\chi(e)$ for every $e\in M$. We construct $\mathbf{B}$ as follows:

1. (1)

The vertices of $\mathbf{B}$ are all pairs $(e,\chi)$ where $e\in M$ and $\chi$ is a function from $M$ to $\{0,1\}$ (called a valuation function). 2. (2)

Distances for $(e,\chi)\neq(f,\chi^{\prime})\in B$ are given by the following rules:

1. (i)

$d_{\mathbf{B}}((e,\chi),(e,1-\chi))=3$, 2. (ii)

$d_{\mathbf{B}}((e,\chi),(f,\chi^{\prime}))=1$ if and only if $\chi(f)=\chi^{\prime}(e)$, 3. (iii)

$d_{\mathbf{B}}((e,\chi),(f,\chi^{\prime}))=2$ otherwise.

Lemma 3.1**.**

The structure $\mathbf{B}$ is an antipodal metric space.

Proof.

Given $(e,\chi)\in B$, by 2(i) we know that there is precisely one vertex at distance 3 (namely $(e,1-\chi)$) and thus the edges of length 3 form a perfect matching.

It remains to check that every quadruple $(e,\chi),(e,1-\chi),(f,\chi^{\prime}),(f,1-\chi^{\prime})$ of distinct vertices of $\mathbf{B}$ is an antipodal metric space. By 2(ii) we know that precisely one of $(f,\chi^{\prime}),(f,1-\chi^{\prime})$ is at distance 1 from $(e,\chi)$ and by 2(iii) that the other is at distance 2, similarly for $(e,1-\chi)$. It also follows that $d_{\mathbf{B}}((e,\chi),(f,\chi^{\prime}))=d_{\mathbf{B}}((e,1-\chi),(f,1-\chi^{\prime}))$ and $d_{\mathbf{B}}((e,\chi),(f,1-\chi^{\prime}))=d_{\mathbf{B}}((e,1-\chi),(f,\chi^{\prime}))$. ∎

We now define an embedding $\psi\colon\mathbf{A}\to\mathbf{B}$ and refer to its image $\mathbf{A}^{\prime}$ in $\mathbf{B}$ as a generic copy of $\mathbf{A}$ in $\mathbf{B}$.

Fix an arbitrary function $p\colon A\to\{0,1\}$ such that whenever $d_{\mathbf{A}}(x,x^{\prime})=3$, then $p(x)=1-p(x^{\prime})$. This function partitions the vertices of $\mathbf{A}$ into two podes such that pairs of vertices at distance 3 are in different podes. For every $1\leq i\leq n$ we denote by $x_{i}$ and $y_{i}$ the endpoints of $e_{i}$ such that $p(x_{i})=0$ and $p(y_{i})=1$. We construct $\psi$ by putting $\psi(x_{i})=(e_{i},\chi_{i})$ and $\psi(y_{i})=(e_{i},1-\chi_{i})$, where $\chi_{i}$ is defined as

[TABLE]

It follows from the construction that $\psi$ is indeed an embedding $\mathbf{A}\to\mathbf{B}$. We put $\mathbf{A}^{\prime}=\psi(\mathbf{A})$. Now we are ready to show the main result of this section:

Proposition 3.2**.**

With the above notation, the antipodal metric space $\mathbf{B}$ is a coherent EPPA-witness for $\mathbf{A}^{\prime}$. Moreover, $p\circ\psi^{-1}$ extends to a function $\hat{p}\colon B\to\{0,1\}$ such that whenever a partial automorphism $\varphi$ of $\mathbf{A}^{\prime}$ preserves values of $p\circ\psi^{-1}$, then its coherent extension $\theta$ preserves values of $\hat{p}$.

Proof.

Let $\varphi$ be a partial automorphism of $\mathbf{A}^{\prime}$. Let $\pi\colon B\to M$ be the projection mapping $(e,\chi)\mapsto e$. By this projection, $\varphi$ induces a partial permutation of $M$ and we denote by $\hat{\varphi}$ an extension of it to a permutation of $M$. To obtain coherence we always extend the permutation in an order-preserving way, that is, we enumerate $M\setminus\mathop{\mathrm{Dom}}\nolimits(\varphi)=\{e_{i_{1}},\ldots,e_{i_{k}}\}$ and $M\setminus\mathop{\mathrm{Range}}\nolimits(\varphi)=\{e_{j_{1}},\ldots,e_{j_{k}}\}$, where $i_{1}<\cdots<i_{k}$ and $j_{1}<\cdots<j_{k}$, and put $\hat{\varphi}(e_{i_{\ell}})=e_{j_{\ell}}$ for every $1\leq\ell\leq k$ (cf.  [SS19, Sol09]).

Let $F$ be the set consisting of unordered pairs $\{e,f\}$, $e,f\in M$ (possibly $e=f$), such that there exists $\chi$ with the property that $(e,\chi)\in\mathop{\mathrm{Dom}}\nolimits(\varphi)$ and $\chi(f)\neq\chi^{\prime}(\hat{\varphi}(f))$ for $(\hat{\varphi}(e),\chi^{\prime})=\varphi((e,\chi))$. We say that these pairs are flipped by $\varphi$. Because of the choice of $\mathbf{A}^{\prime}$, there are zero or two choices for $\chi$ for every $e$, and if there are two, then they are $\chi$ and $1-\chi$ for some $\chi$ and both of them give the same result. Note that there may be no $\eta$ such that $(f,\eta)\in\mathop{\mathrm{Dom}}\nolimits(\varphi)$.

Define a function $\theta\colon B\to B$ by putting

[TABLE]

where

[TABLE]

First we verify that $\theta$ extends $\varphi$. Suppose $(e,\chi)\in\mathop{\mathrm{Dom}}\nolimits(\varphi)$. Write $\theta(e,\chi)=(\hat{\varphi}(e),\xi)$ and $\varphi(e,\chi)=(\hat{\varphi}(e),\chi^{\prime})$. We must check that $\xi=\chi^{\prime}$, so we let $f\in M$ and show that $\chi^{\prime}(\hat{\varphi}(f))=\xi(\hat{\varphi}(f))$. But this follows easily from the definitions of $\xi$ and $F$, by considering the cases $\{e,f\}\in F$ and $\{e,f\}\not\in F$ separately.

Now we check that $\theta$ is an automorphism of $\mathbf{B}$. It is easy to see that $\theta$ is one-to-one (one can construct its inverse) and that it preserves antipodal pairs. To check that $d_{\mathbf{B}}((e,\chi),(f,\eta))=d_{\mathbf{B}}(\theta((e,\chi)),\theta((f,\eta)))$ for non-antipodal pairs, denote $\theta((e,\chi))=(\hat{\varphi}(e),\chi^{\prime})$ and $\theta((f,\eta))=(\hat{\varphi}(f),\eta^{\prime})$.

By definition of $\theta$, we have $\chi(f)\neq\chi^{\prime}(\hat{\varphi}(f))$ if and only if $\{e,f\}\in F$, and analogously, $\eta(e)\neq\eta^{\prime}(\hat{\varphi}(e))$ if and only if $\{e,f\}\in F$. Putting this together (and again considering the cases $\{e,f\}\in F$ and $\{e,f\}\not\in F$ separately) we get $\chi(f)=\eta(e)$ if and only if $\chi^{\prime}(\hat{\varphi}(f))=\eta^{\prime}(\hat{\varphi}(e))$. Together with the definition of $d_{\mathbf{B}}$ this implies that indeed $d_{\mathbf{B}}((e,\chi),(f,\eta))=d_{\mathbf{B}}(\theta((e,\chi)),\theta((f,\eta)))$.

Thus far, we have shown that $\mathbf{B}$ is an EPPA-witness for $\mathbf{A}^{\prime}$.

Now we put $\hat{p}((e,\chi))=\chi(e)$. Note that $\hat{p}(\psi(x_{i}))=\chi_{i}(e_{i})=0=p(x_{i})$ and similarly $\hat{p}(\psi(y_{i}))=p(y_{i})$. So $\hat{p}$ extends $p\circ\psi^{-1}$.

Suppose $\varphi$ preserves values of $p$. Then for $(e,\chi)\in\mathop{\mathrm{Dom}}\nolimits(\varphi)$ it holds that $\chi(e)=\chi^{\prime}(\hat{\varphi}(e))$, where $(\hat{\varphi}(e),\chi^{\prime})=\varphi((e,\chi))$. Thus there is no $e\in M$ such that $\{e\}\in F$ (for such an $e$, there would have to be some $(e,\chi)\in\mathop{\mathrm{Dom}}\nolimits\varphi$, of course). By definition of $\theta$ we immediately get $\hat{p}((e,\chi))=\hat{p}(\theta((e,\chi)))$ for every $(e,\chi)\in B$.

Finally we verify coherence for the above construction. Consider partial automorphisms $\varphi_{1}$, $\varphi_{2}$ and $\varphi$ of $\mathbf{A}^{\prime}$ such that $\varphi$ is the composition of $\varphi_{1}$ and $\varphi_{2}$. Denote by $\hat{\varphi}_{1}$, $\hat{\varphi}_{2}$ and $\hat{\varphi}$ their corresponding permutations of $M$ constructed above. Let $F_{1}$, $F_{2}$ and $F$ be the corresponding sets of flipped pairs and $\theta_{1}$, $\theta_{2}$ and $\theta$ the corresponding extensions. Because the permutations $\hat{\varphi}_{1}$, $\hat{\varphi}_{2}$ and $\hat{\varphi}$ were constructed by extending projections of $\varphi_{1}$, $\varphi_{2}$ and $\varphi$ (which also compose coherently) in an order-preserving way, we know that $\hat{\varphi}$ is the composition of $\hat{\varphi}_{1}$ and $\hat{\varphi}_{2}$.

To see that $\theta$ is the composition of $\theta_{1}$ and $\theta_{2}$ one first checks that for $e\in\pi(\mathop{\mathrm{Dom}}\nolimits(\varphi))=\pi(\mathop{\mathrm{Dom}}\nolimits(\varphi_{1}))$ and $f\in M$ one has that $\{e,f\}\in F$ if and only if

[TABLE]

The definition of $\theta_{1},\theta_{2}$ and $\theta$ then gives the required result. In more detail, write $\theta_{1}(e,\chi)=(\hat{\varphi}_{1}(e),\xi_{1})$ and $\theta_{2}\theta_{1}((e,\chi))=\theta_{2}(\hat{\varphi}_{1}(e),\xi_{1})=(\hat{\varphi}(e),\xi_{2})$, where, for $f\in M$,

[TABLE]

Applying the definition of $\xi_{1}$ and using the above observation finishes the calculation.

∎

4. Proofs of the main results

Theorem 1.3 is a direct consequence of Proposition 3.2. Next we use the correspondence between graphs with switching automorphisms and antipodal metric spaces (or, in the language of Seidel, double-covers of complete graphs [Cam99, Sei73]) to prove Theorem 1.2:

Proof of Theorem 1.2.

Given a graph $\mathbf{G}$, we construct an antipodal metric space $\mathbf{A}$ on vertex set $G\times\{0,1\}$ with distances defined as follows:

1. (1)

$d_{\mathbf{A}}((x,0),(x,1))=3$ for every $x\in G$, 2. (2)

$d_{\mathbf{A}}((x,i),(y,i))=1$ for every $x\neq y$ forming an edge of $\mathbf{G}$ and $i\in\{0,1\}$, 3. (3)

$d_{\mathbf{A}}((x,i),(y,1-i))=1$ for every $x\neq y$ forming a non-edge of $\mathbf{G}$ and $i\in\{0,1\}$, and 4. (4)

$d_{\mathbf{A}}((x,i),(y,j))=2$ otherwise.

Let $p\colon A\to\{0,1\}$ be a function defined by $p((x,i))=i$. We apply Proposition 3.2 to construct an antipodal metric space $\mathbf{C}$ and a function $\hat{p}\colon C\to\{0,1\}$. Construct a graph $\mathbf{H}$ with vertex set $\{x\in C:\hat{p}(x)=0\}$ with $x,y$ forming an edge if and only if $d_{\mathbf{C}}(x,y)=1$.

Now consider a partial automorphism $\varphi$ of $\mathbf{G}$. This automorphism corresponds to a partial automorphism $\varphi^{\prime}$ of $\mathbf{A}$ by putting $\varphi^{\prime}((x,i))=(\varphi(x),i)$ for every $x\in\mathop{\mathrm{Dom}}\nolimits(\varphi)$ and $i\in\{0,1\}$. $\varphi^{\prime}$ then extends to $\theta$ which preserves values of $\hat{p}$. Consequently, $\theta$ restricted to $H$ is an automorphism of $\mathbf{H}$.

Finally consider a partial switching automorphism $\varphi$ (i.e. a switching isomorphism of induced subgraphs). Let $S$ be the set of vertices switched by $\varphi$. Now the partial automorphism of $\mathbf{A}$ is defined by putting $\varphi^{\prime}((x,i))=(\varphi(x),i)$ if $x\notin S$ and $\varphi^{\prime}((x,i))=(\varphi(x),1-i)$ otherwise. Again extend $\varphi^{\prime}$ to $\theta$ and observe that $\theta$ gives a switching automorphism of $\mathbf{H}$. ∎

EPPA for two-graphs follows easily too:

Proof of Theorem 1.1.

Let $\mathbf{T}$ be a finite two-graph, pick an arbitrary vertex $x\in T$ and define a graph $\mathbf{G}$ on the vertex set $T$ such that $\{y,z\}\in E_{\mathbf{G}}$ if and only if $x\notin\{y,z\}$ and $\{x,y,z\}$ is a triple of $\mathbf{T}$. Observe that $\mathbf{T}=T(\mathbf{G})$. By Theorem 1.2, there is a graph $\mathbf{H}$ containing $\mathbf{G}$ such that every switching isomorphism of induced subgraphs of $\mathbf{G}$ extends to a switching automorphism of $\mathbf{H}$. We claim that $T(\mathbf{H})$ is an EPPA-witness for $\mathbf{T}$.

To prove that, let $\varphi\colon\mathbf{T}\to\mathbf{T}$ be a partial automorphism of $\mathbf{T}$. By the construction of $\mathbf{G}$ and the correspondence between graphs and two-graphs, $\varphi$ is a switching isomorphism of subgraphs of $\mathbf{G}$ induced on the domain and range of $\varphi$ respectively. Hence, it extends to a switching automorphism of $\mathbf{H}$ and thus an automorphism of $T(\mathbf{H})$, which is what we wanted. ∎

Remark 4.1**.**

Observe that the EPPA-witness given in this proof of Theorem 1.1 is not necessarily a coherent EPPA-witness. The problem lies in the proof of Theorem 1.2, because for every switching isomorphism of subgraphs of $\mathbf{G}$ there are two corresponding partial automorphisms of $\mathbf{A}$. For example, if the partial automorphism of $\mathbf{G}$ is a partial identity, one partial automorphism of $\mathbf{A}$ is also a partial identity, while the other flips all the involved edges of length 3. While the first is extended to the identity by the construction in Proposition 3.2, the other is extended to a non-trivial permutation of the edges of length 3. This issue carries over to Theorem 1.1, and in fact, it seems to be a fundamental obstacle for using antipodal metric spaces to prove coherent EPPA for two-graphs.

5. Existence of a dense locally finite subgroup

As discussed in Section 2.1 here, Solecki and Siniora [SS19, Sol09] introduced the notion of coherent EPPA for a Fraïssé class as a way to prove that the automorphism group of the respective Fraïssé limit contains a dense locally finite subgroup. While we cannot prove coherent EPPA for $\mathcal{T}$, Theorem 1.3 gives coherent EPPA for the class of all antipodal metric spaces of diameter 3 and thus there is a dense locally finite subgroup of the automorphism group of its Fraïssé limit. We now show how this gives a dense, locally finite subgroup of the automorphism group of the Fraïssé limit of $\mathcal{T}$.

Let $\mathbb{T}$ be the Fraïssé limit of $\mathcal{T}$ and $\mathbb{M}$ be the Fraïssé limit of the class of all finite antipodal metric spaces of diameter $3$. The main result of this section is the following theorem.

Theorem 5.1**.**

There is a dense, locally finite subgroup of $\mathop{\mathrm{Aut}}\nolimits(\mathbb{T})$.

In order to prove this, we first describe how to get a two-graph from an antipodal metric space, which is again a well-known construction [Cam99, Sei73].

In an antipodal metric space $\mathbf{A}$, every quadruple of distinct vertices $a,b,c,d$ such that $d_{\mathbf{A}}(a,b)=d_{\mathbf{A}}(c,d)=3$ (a pair of edges of label 3) induces one of the two (isomorphic) subspaces depicted in Figure 1 — we call these antipodal quadruples. However, three edges with label $3$ can induce two non-isomorphic structures; the edges of length 1 either form two triangles, or one 6-cycle (see Figure 2). This motivates the following correspondence:

Definition 5.1** (From antipodal spaces to two-graphs).**

Let $\mathbf{A}$ be an antipodal metric space. Let $M$ be the set of all edges of $\mathbf{A}$ of length 3 (thus, $|M|=\frac{|A|}{2}$ if $\mathbf{A}$ is finite). Define $T(\mathbf{A})$ to be the 3-uniform hypergraph on vertex set $M$ where $\{a,b,c\}$ is a hyperedge if and only if the substructure of $\mathbf{A}$ induced on the edges $a,b,c$ is isomorphic to that depicted in Figure 2 (b).

It is straightforward to verify that $T(\mathbf{A})$ is a two-graph. Clearly there is a natural two-to-one map $\mathbf{A}\to T(\mathbf{A})$ and this induces a group homomorphism $\mathop{\mathrm{Aut}}\nolimits(\mathbf{A})\to\mathop{\mathrm{Aut}}\nolimits(T(\mathbf{A}))$. It is also well-known that there is a converse to this construction (usually expressed in terms of double covers of compete graphs). Suppose $\mathbf{T}$ is a two-graph. Let $\mathbf{G}$ be a graph in the switching class of $\mathbf{T}$ (so $T(\mathbf{G})=\mathbf{T}$) and let $\mathbf{A}$ be the antipodal metric space constructed in the proof of Theorem 1.2. Then $T(\mathbf{A})=\mathbf{T}$ and the proof of Theorem 1.2 shows that the map $\mathop{\mathrm{Aut}}\nolimits(\mathbf{A})\to\mathop{\mathrm{Aut}}\nolimits(\mathbf{T})$ is surjective, as any automorphism of $\mathbf{T}$ is a switching automorphism of $\mathbf{G}$. More generally, a similar argument gives the following well-known fact:

Lemma 5.2**.**

If $\mathbf{A}_{1},\mathbf{A}_{2}$ are antipodal metric spaces and $\beta\colon T(\mathbf{A}_{1})\to T(\mathbf{A}_{2})$ is an isomorphism of two-graphs, then there is an isomorphism $\alpha\colon\mathbf{A}_{1}\to\mathbf{A}_{2}$ which induces $\beta$.

We can now give:

Proof of Theorem 5.1.

Let $\mathbf{T}$ be the two-graph $T(\mathbb{M})$. By Lemma 5.2, any isomorphism between finite substructures of $\mathbf{T}$ lifts to an isomorphism between finite substructures of $\mathbb{M}$ and so, as $\mathbb{M}$ is homogeneous, this partial isomorphism is induced by an automorphism of $\mathbb{M}$. This shows that $\mathbf{T}$ is homogeneous. The construction of an antipodal metric space from a two-graph shows that $\mathbf{T}$ embeds every finite two-graph, and therefore $\mathbf{T}$ is isomorphic to $\mathbb{T}$. So we have, again using Lemma 5.2, a surjective homomorphism $\alpha\colon\mathop{\mathrm{Aut}}\nolimits(\mathbb{M})\to\mathop{\mathrm{Aut}}\nolimits(\mathbb{T})$. As already observed, there is a dense, locally finite subgroup $H$ of $\mathop{\mathrm{Aut}}\nolimits(\mathbb{M})$. Then $\alpha(H)$ is a dense, locally finite subgroup of $\mathop{\mathrm{Aut}}\nolimits(\mathbb{T})$.

∎

6. Amalgamation property with automorphisms

Let $\mathbf{A}$, $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ be structures, $\alpha_{1}$ an embedding of $\mathbf{A}$ into $\mathbf{B}_{1}$ and $\alpha_{2}$ an embedding of $\mathbf{A}$ into $\mathbf{B}_{2}$. Then every structure $\mathbf{C}$ with embeddings $\beta_{1}\colon\mathbf{B}_{1}\to\mathbf{C}$ and $\beta_{2}\colon\mathbf{B}_{2}\to\mathbf{C}$ such that $\beta_{1}\circ\alpha_{1}=\beta_{2}\circ\alpha_{2}$ is called an amalgamation of $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ over $\mathbf{A}$ with respect to $\alpha_{1}$ and $\alpha_{2}$. Amalgamation is strong if $\beta_{1}(x_{1})=\beta_{2}(x_{2})$ if and only if $x_{1}\in\alpha_{1}(A)$ and $x_{2}\in\alpha_{2}(A)$.

For simplicity, in the following definition we will assume that all the embeddings in the definition of amalgamation are in fact inclusions.

Definition 6.1** (APA).**

Let $\mathcal{C}$ be a class of finite structures. We say that $\mathcal{C}$ has the amalgamation property with automorphisms (APA) if for every $\mathbf{A},\mathbf{B}_{1},\mathbf{B}_{2}\in\mathcal{C}$ such that $\mathbf{A}\subseteq\mathbf{B}_{1},\mathbf{B}_{2}$ there exists $\mathbf{C}\in\mathcal{C}$ which is an amalgamation of $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ over $\mathbf{A}$, has $\mathbf{B}_{1},\mathbf{B}_{2}\subseteq\mathbf{C}$ and furthermore the following holds:

For every pair of automorphisms $f_{1}\in\mathop{\mathrm{Aut}}\nolimits(\mathbf{B}_{1})$, $f_{2}\in\mathop{\mathrm{Aut}}\nolimits(\mathbf{B}_{2})$ such that $f_{i}(A)=A$ for $i\in\{1,2\}$ and $f_{1}|_{A}=f_{2}|_{A}$, there is an automorphism $g\in\mathop{\mathrm{Aut}}\nolimits(\mathbf{C})$ such that $g|_{B_{i}}=f_{i}$ for $i\in\{1,2\}$.

In other words, APA is a strengthening of the amalgamation property which requires that every pair of automorphisms of $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ which agree on $\mathbf{A}$ extends to an automorphism of $\mathbf{C}$.

As was mentioned in the introduction, EPPA + APA imply the existence of ample generics [HHLS93] and it turns out that most of the known EPPA classes also have APA. We now show that this is not the case for two-graphs.

Proposition 6.1**.**

Let $\mathbf{A}$ be the two-graph on two vertices, $\mathbf{B}_{1}$ be the two-graph on three vertices with no hyper-edge and $\mathbf{B}_{2}$ be the two-graph on three vertices which form a hyper-edge. It is possible to amalgamate $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ over $\mathbf{A}$, but it is not possible to amalgamate them with automorphisms.

Proof.

For convenience, we name the vertices as in Figure 3: $A=\{u,v\}$, $B_{1}=\{u,v,x_{1}\}$ and $B_{2}=\{u,v,x_{2}\}$. For $i\in\{1,2\}$ let $f_{i}$ be the automorphism of $\mathbf{B}_{i}$ such that $f_{i}(x_{i})=x_{i}$, $f_{i}(u)=v$ and $f_{i}(v)=u$. Clearly $f_{1}$ and $f_{2}$ agree on $\mathbf{A}$.

Consider the amalgamation problem for $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ over $\mathbf{A}$ (with inclusions and with automorphisms $f_{1},f_{2}$) and assume for a contradiction that there is $\mathbf{C}\in\mathcal{T}$ and an automorphism $g$ of $\mathbf{C}$ as in Definition 6.1. By the definition of $\mathcal{T}$, we get that there has to be an even number of triples in $\mathbf{C}$ on $\{u,v,x_{1},x_{2}\}$ and since we know that $\{u,v,x_{1}\}$ is not a triple and $\{u,v,x_{2}\}$ is a triple, there actually have to be precisely two triples on $\{u,v,x_{1},x_{2}\}$. Therefore, exactly one of $\{u,x_{1},x_{2}\}$ and $\{v,x_{1},x_{2}\}$ has to form a triple in $\mathbf{C}$. But this means that $g$ is not an automorphism (as $g$ fixes $x_{1}$ and $x_{2}$ and swaps $u$ and $v$), a contradiction.

On the other hand, if we only want to amalgamate $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ over $\mathbf{A}$ (not with automorphisms), then this is clearly possible. ∎

Remark 6.1**.**

In the introduction we mentioned that there are also some pathological examples with EPPA but not APA:

1. (1)

Let $\mathbf{M}$ be the two-vertex set with no structure. Its age consists of the empty set, the one-element set and $\mathbf{M}$. Consider the amalgamation problem for $\mathbf{A}$ the empty set and $\mathbf{B}_{1}=\mathbf{B}_{2}=\mathbf{M}$. The amalgam has to be $\mathbf{M}$ again. But then it is impossible to preserve all four possible pairs of automorphisms of $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$.

This phenomenon clearly happens because this is not a strong amalgamation class (and is exhibited by other non-strong amalgamation classes) and, indeed, disappears when we only consider closed structures. 2. (2)

Let now $\mathbf{M}$ be the disjoint union of two infinite cliques, that is, an equivalence relation with two equivalence classes and consider its age. Let $\mathbf{A}$ be the empty structure and $\mathbf{B}_{1}=\mathbf{B}_{2}$ consist of two non-equivalent vertices. This amalgamation problem, again, does not have an APA-solution, because one needs to decide which pairs of vertices will be equivalent and this cannot preserve all four pairs of automorphisms.

This generalises to situations where the homogeneous structure has a definable equivalence relation with finitely many equivalence classes (or, more generally, of finite index).

However, the reason here is that the equivalence classes are algebraic in a quotient, i.e. a similar reason as in the previous point. One can either require the amalgamation problem to specify which classes go to which ones or, equivalently, consider an expansion which weakly eliminates imaginaries and the problem disappears.

While these two examples point out that, at least from the combinatorial point of view, one needs a more robust definition for APA, two-graphs seem to innately not have APA.

7. Ramsey expansion of two-graphs

As was pointed out recently, the methods for proving EPPA and the Ramsey property share many similarities, see e.g. [ABWH*+*17]. The standard strategy is to study the completion problem for given classes and their expansions, see [ABWH*+*17, EHN17, Kon19a, HKN18]. EPPA is usually a corollary of one of the steps towards finding a Ramsey expansion.

The situation is very different for two-graphs. As we shall observe in this section, the Ramsey question can be easily solved using standard techniques, while for EPPA this is not the case (see Remark 8.1). This makes two-graphs an important example of the limits of the current methods and shows that the novel approach of this paper is in fact necessary.

We now give the very basic definitions of the structural Ramsey theory.

A class ${\mathcal{K}}$ of structures is Ramsey if for every $\mathbf{A},\mathbf{B}\in{\mathcal{K}}$ there exists $\mathbf{C}\in{\mathcal{K}}$ such that for every colouring of copies of $\mathbf{A}$ in $\mathbf{C}$ there exists a copy of $\mathbf{B}$ in $\mathbf{C}$ that is monochromatic. (By a copy of $\mathbf{A}$ in $\mathbf{C}$ we mean any substructure of $\mathbf{C}$ isomorphic to $\mathbf{A}$.) This statement is abbreviated as $\mathbf{C}\longrightarrow(\mathbf{B})_{k}^{\mathbf{A}}$.

If class $\mathcal{K}$ has joint embedding property (which means that for every $\mathbf{A}_{1},\mathbf{A}_{2}\in\mathcal{K}$ there is $\mathbf{C}\in\mathcal{K}$ which contains a copy of both $\mathbf{A}_{1}$ and $\mathbf{A}_{2}$) then EPPA of $\mathcal{K}$ implies amalgamation. It is well known that Ramsey property also implies amalgamation property [Neš89, Neš05] under the assumption of joint embedding.

Every Ramsey class consists of rigid structures, i.e. structures with no non-trivial automorphism. The usual way to establish rigidity is to extend the language (in a model-theoretical way) by an additional binary relation $\leq$ which fixes the ordering of vertices. It is thus a natural question whether the class $\overrightarrow{\mathcal{T}}$ of all two-graphs with a linear ordering of the vertices is Ramsey. We now show that the answer is negative and to do it, we need the following statement.

Proposition 7.1**.**

For every two-graph $\mathbf{A}$ there exists a two-graph $\mathbf{B}$ such that every graph in the switching class of $\mathbf{B}$ contains a copy of every graph in the switching class of $\mathbf{A}$.

Proof.

Denote by $\mathbf{G}$ the disjoint union of all graphs in the switching class of $\mathbf{A}$. Now let $\mathbf{H}$ be a graph such that every colouring of vertices of $\mathbf{H}$ by 2 colours induces a monochromatic copy of $\mathbf{G}$ (that is, $\mathbf{H}$ is vertex-Ramsey for $\mathbf{G}$) — it exists by a theorem of Folkman [Fol70, NR76, NR77a]).

Every graph $\mathbf{H}^{\prime}$ in the switching class of $\mathbf{H}$ induces a colouring of vertices of $\mathbf{H}$ by two colours: $\mathbf{H}^{\prime}$ being in the switching class of $\mathbf{H}$ means that there is a set $S\subseteq H$ such that $\mathbf{H}^{\prime}=\mathbf{H}_{S}$, the colour classes are then $S$ and $H\setminus S$ respectively.

By the construction of $\mathbf{H}$ we find a copy $\widetilde{\mathbf{G}}\subseteq\mathbf{H}$ of $\mathbf{G}$ which is monochromatic with respect to this colouring. This however implies that the graphs induced by $\mathbf{H}$ and $\mathbf{H}^{\prime}$ on the set $\widetilde{G}$ are isomorphic and thus $\mathbf{H}^{\prime}$ indeed contains every graph in the switching class of $\mathbf{A}$. Therefore, we can put $\mathbf{B}$ to be the two-graph associated to $\mathbf{H}$. ∎

Corollary 7.2**.**

The class $\overrightarrow{\mathcal{T}}$ is not Ramsey for colouring pairs of vertices.

Proof.

Let $\mathbf{A}$ be the two-graph associated to an arbitrary graph containing both an edge and a non-edge, and let $\mathbf{B}$ be the two-graph given by Proposition 7.1 for $\mathbf{A}$. Let $\overrightarrow{\mathbf{B}}$ be an arbitrary linear ordering of $\mathbf{B}$.

Assume that there exists an ordered two-graph $\overrightarrow{\mathbf{C}}$ such that

[TABLE]

where $\overrightarrow{\mathbf{E}}$ is the unique ordered two-graph on 2 vertices.

Let $\overrightarrow{\mathbf{I}}$ be an arbitrary graph from the switching class of $\overrightarrow{\mathbf{C}}$ (with the inherited order) and colour copies of $\overrightarrow{\mathbf{E}}$ red if they correspond to an edge of $\overrightarrow{\mathbf{I}}$ and blue otherwise. By the construction of $\mathbf{B}$ it follows that there is no monochromatic copy of $\overrightarrow{\mathbf{B}}$, a contradiction. ∎

Proposition 7.1 says that the expansion of two-graphs adding a particular graph from the switching class has the so-called expansion property. As a consequence of the Kechris–Pestov–Todorčević correspondence [KPT05], one gets that every Ramsey expansion of $\mathcal{T}$ has to fix a particular representative of the switching class (see e.g. [NVT15] for details, it is also easy to see this directly). On the other hand, expanding any two-graph by a linear order and a particular graph from the given switching class is a Ramsey expansion by the Nešetřil-Rödl theorem [NR77b].

8. Remarks

Remark 8.1**.**

The by-now-standard strategy for proving EPPA for class $\mathcal{C}$ in, say, a relational language $L$ may be summarized as follows:

1. (1)

Assume that, every pair of vertices of every structure in $\mathcal{C}$ is in some relation. If it is not, we can always add a new binary relation to $L$ and put all pairs into the relation. 2. (2)

Study the class $\mathcal{C}^{-}$ which consists of all $L$-structures $\mathbf{A}^{-}$ such that there is $\mathbf{A}\in\mathcal{C}$ which is a completion of $\mathbf{A}^{-}$, that is, $\mathbf{A}^{-}$ and $\mathbf{A}$ have the same vertex set $A$ and there is $X\subset\mathcal{P}(A)$, a subset of the powerset of $A$, such that if $Y\in X$ and $Z\subseteq Y$, then $Z\in X$, and for each relation $R\in L$ it holds that $R^{\mathbf{A}^{-}}=R^{\mathbf{A}}\cap X$. 3. (3)

Find a finite family of $L$-structures $\mathcal{F}$ such that $\mathcal{C}^{-}$ is precisely $\mathop{\mathrm{Forb}}\nolimits(\mathcal{F})$, that is, the class of all finite $L$-structures $\mathbf{B}$ such that there is no $\mathbf{F}\in\mathcal{F}$ with a homomorphism to $\mathbf{B}$. 4. (4)

Prove that in fact for every $\mathbf{A}^{-}\in\mathcal{C}^{-}$ there is $\mathbf{A}\in\mathcal{C}$ which is its automorphism-preserving completion, that is, $\mathbf{A}^{-}$ can be obtained from $\mathbf{A}$ as in point 2 and furthermore $\mathbf{A}^{-}$ and $\mathbf{A}$ have the same automorphisms. 5. (5)

Use the Herwig–Lascar theorem [HL00] omitting homomorphisms from $\mathcal{F}$ to get EPPA-witnesses in $\mathop{\mathrm{Forb}}\nolimits(\mathcal{F})$. 6. (6)

Take the automorphism-preserving completion of the witnesses to get EPPA-witnesses in $\mathcal{C}$ and thus prove EPPA for $\mathcal{C}$.

In various forms, this strategy was applied, for example, in [Sol05, Con19, ABWH*+*17, Kon19a, HKN18]. See also [HN19] where the notion of completions was introduced.

As we have seen in Section 6, $\mathcal{T}$ does not admit automorphism-preserving completions (because APA is a weaker property). One can also prove (and it will appear elsewhere), using the negative result of Proposition 7.1 and Theorem 2.11 of [HN19], that $\mathcal{T}$ cannot be described by finitely many forbidden homomorphisms (hence in particular there is no finite family $\mathcal{F}$ satisfying point 3 above). This we believe is the first time the Ramsey techniques have been used to prove a negative result for EPPA.

Remark 8.2**.**

$\mathcal{T}$ is one of the five reducts of the random graph [Tho91]. Besides $\mathcal{T}$, the random graph itself and the countable set with no structure, the remaining two corresponding automorphism groups can be obtained by adding an isomorphism between the random graph and its complement and an isomorphism between the generic two-graph and its complement respectively.

By a similar argument, one can prove that the “best” Ramsey expansion (that is, with the expansion property) of these structures is still the ordered random graph. On the other hand, EPPA for these two classes is an open problem (we conjecture that neither of these two classes has EPPA).

Remark 8.3**.**

Theorem 1.2 implies the following. For every graph $\mathbf{G}$ there exists an EPPA-witness $\mathbf{H}$ with the property that the two-graph associated to $\mathbf{H}$ is an EPPA-witness for the two-graph associated for $\mathbf{G}$, in other words, it implies that the class of all graphs and two-graphs respectively are a non-trivial positive example for the following question.

Question 8.1**.**

For which pairs of classes $\mathcal{C}$, $\mathcal{C}^{-}$ such that $\mathcal{C}^{-}$ is a reduct of $\mathcal{C}$ does it hold that for every $\mathbf{A}\in\mathcal{C}$ there is $\mathbf{B}\in\mathcal{C}$ such that $\mathbf{B}$ is an EPPA-witness for $\mathbf{A}$ (in $\mathcal{C}$) and furthermore if $\mathbf{A}^{-}$ and $\mathbf{B}^{-}$ are the corresponding reducts in $\mathcal{C}^{-}$ then $\mathbf{B}^{-}$ is an EPPA-witness for $\mathbf{A}^{-}$ (in $\mathcal{C}^{-}$)?

Remark 8.4**.**

As was already mentioned, ample generics are usually proved to exist by showing the combination of EPPA and APA. Siniora in his thesis [Sin17] asked if two-graphs have ample generics. This question still remains open, although we conjecture that it is not the case (ample generics are equivalent to having the so-called weak amalgamation property for partial automorphisms and the joint embedding property for partial automorphisms and it seems that the reasons for two-graphs not having APA are fundamental enough to also hold in the weak amalgamation context).111Added August 2019: in joint work with P. Simon, the authors have shown that this conjecture holds.

Acknowledgment

We would like to thank the anonymous referee for their helpful comments.