On Hrushovski properties of Hrushovski constructions Jan Hubi cka - - PowerPoint PPT Presentation

EPPA C0 CF Summary

On Hrushovski properties of Hrushovski constructions

Jan Hubiˇ cka

Department of Applied Mathematics Charles University Prague Joint work with David Evans, Matˇ ej Koneˇ cný, and Jaroslav Nešetˇ ril

Logic Colloquium 2019, Prague

EPPA C0 CF Summary

Definition (Extension property for partial automorphisms) A class C of finite L-structures has extension property for partial automorphisms (EPPA

• r Hrushovski property) iff for every A ∈ C

EPPA C0 CF Summary

Definition (Extension property for partial automorphisms) A class C of finite L-structures has extension property for partial automorphisms (EPPA

• r Hrushovski property) iff for every A ∈ C

A

EPPA C0 CF Summary

Definition (Extension property for partial automorphisms) A class C of finite L-structures has extension property for partial automorphisms (EPPA

• r Hrushovski property) iff for every A ∈ C there exists EPPA witness B ∈ C containing

A

A B

EPPA C0 CF Summary

Definition (Extension property for partial automorphisms) A class C of finite L-structures has extension property for partial automorphisms (EPPA

• r Hrushovski property) iff for every A ∈ C there exists EPPA witness B ∈ C containing

A such that every partial automorphism of A Partial automorphism is any isomorphism between two substructures.

A B

EPPA C0 CF Summary

Definition (Extension property for partial automorphisms) A class C of finite L-structures has extension property for partial automorphisms (EPPA

• r Hrushovski property) iff for every A ∈ C there exists EPPA witness B ∈ C containing

A such that every partial automorphism of A extends to automorphism of B. Partial automorphism is any isomorphism between two substructures.

A B

EPPA C0 CF Summary

Definition (Extension property for partial automorphisms) A class C of finite L-structures has extension property for partial automorphisms (EPPA

• r Hrushovski property) iff for every A ∈ C there exists EPPA witness B ∈ C containing

A such that every partial automorphism of A extends to automorphism of B. Partial automorphism is any isomorphism between two substructures. Example (Classes with EPPA)

1 Graphs (Hrushovski 1992) 2 Relational structures (Herwig 1998) 3 Classes described by finite forbidden homomorphisms (Herwig-lascar 2000) 4 Free amalgamation classes (Hodkinson and Otto 2003) 5 Metric spaces (Solecki 2005, Vershik 2008) 6 Generalisations and specialisations of metric spaces (Conant 2015)

EPPA C0 CF Summary

Definition (Extension property for partial automorphisms) A class C of finite L-structures has extension property for partial automorphisms (EPPA

• r Hrushovski property) iff for every A ∈ C there exists EPPA witness B ∈ C containing

A such that every partial automorphism of A extends to automorphism of B. Partial automorphism is any isomorphism between two substructures. Example (Classes with EPPA)

1 Graphs (Hrushovski 1992)

Ramsey with free linear order (Nešetˇ ril-Rödl 1977, Abramson-Harrington 1978)

2 Relational structures (Herwig 1998)

Ramsey with free linear order (N. R. 1977, A.H. 1978)

3 Classes described by finite forbidden homomorphisms (Herwig-lascar 2000)

Ramsey with free linear order (H.-Nešetˇ ril 2016)

4 Free amalgamation classes (Hodkinson and Otto 2003)

Ramsey with free linear order (Nešetˇ ril-Rödl 1977)

5 Metric spaces (Solecki 2005, Vershik 2008)

Ramsey with free linear order (Nešetˇ ril 2005)

6 Generalisations and specialisations of metric spaces (Conant 2015)

Ramsey with convex linear order (Nguyen Van Thé 2010, H.-Nešetˇ ril 2016)

EPPA C0 CF Summary

Hrushovski (predimension) construction

• Predimension of a graph G = (V, E) is δ(G) = 2|V| − |E|.

Example δ(K1) = 2 δ(K2) = 4 − 1 = 3 δ(K3) = 6 − 3 = 3 δ(K4) = 8 − 6 = 2 δ(K5) = 10 − 10 = 0 δ(K6) = 12 − 30 = −18.

EPPA C0 CF Summary

Hrushovski (predimension) construction

• Predimension of a graph G = (V, E) is δ(G) = 2|V| − |E|.

Example δ(K1) = 2 δ(K2) = 4 − 1 = 3 δ(K3) = 6 − 3 = 3 δ(K4) = 8 − 6 = 2 δ(K5) = 10 − 10 = 0 δ(K6) = 12 − 30 = −18.

• Finite graph G is in C0 iff ∀H⊆Gδ(H) ≥ 0.

EPPA C0 CF Summary

Hrushovski (predimension) construction

• Predimension of a graph G = (V, E) is δ(G) = 2|V| − |E|.

Example δ(K1) = 2 δ(K2) = 4 − 1 = 3 δ(K3) = 6 − 3 = 3 δ(K4) = 8 − 6 = 2 δ(K5) = 10 − 10 = 0 δ(K6) = 12 − 30 = −18.

• Finite graph G is in C0 iff ∀H⊆Gδ(H) ≥ 0.
• G ⊆ H is self-sufficient iff ∀G⊆G′⊆Hδ(G) ≤ δ(G′).

EPPA C0 CF Summary

Hrushovski (predimension) construction

• Predimension of a graph G = (V, E) is δ(G) = 2|V| − |E|.

Example δ(K1) = 2 δ(K2) = 4 − 1 = 3 δ(K3) = 6 − 3 = 3 δ(K4) = 8 − 6 = 2 δ(K5) = 10 − 10 = 0 δ(K6) = 12 − 30 = −18.

• Finite graph G is in C0 iff ∀H⊆Gδ(H) ≥ 0.
• G ⊆ H is self-sufficient iff ∀G⊆G′⊆Hδ(G) ≤ δ(G′).

Definition (Amalgamation property of class K)

A B B′ C

EPPA C0 CF Summary

Hrushovski (predimension) construction

• Predimension of a graph G = (V, E) is δ(G) = 2|V| − |E|.

Example δ(K1) = 2 δ(K2) = 4 − 1 = 3 δ(K3) = 6 − 3 = 3 δ(K4) = 8 − 6 = 2 δ(K5) = 10 − 10 = 0 δ(K6) = 12 − 30 = −18.

• Finite graph G is in C0 iff ∀H⊆Gδ(H) ≥ 0.
• G ⊆ H is self-sufficient iff ∀G⊆G′⊆Hδ(G) ≤ δ(G′).

Definition (Amalgamation property of class K)

A B B′ C

Lemma C0 is closed for free amalgamation over self-sufficient substructures. Proof. δ(C) = δ(B) + δ(B′) − δ(A).

A B B′ C

EPPA C0 CF Summary

Hrushovski (predimension) construction

• Predimension of a graph G = (V, E) is δ(G) = 2|V| − |E|.

Example δ(K1) = 2 δ(K2) = 4 − 1 = 3 δ(K3) = 6 − 3 = 3 δ(K4) = 8 − 6 = 2 δ(K5) = 10 − 10 = 0 δ(K6) = 12 − 30 = −18.

• Finite graph G is in C0 iff ∀H⊆Gδ(H) ≥ 0.
• G ⊆ H is self-sufficient iff ∀G⊆G′⊆Hδ(G) ≤ δ(G′).

Definition (Amalgamation property of class K)

A B B′ C

Lemma C0 is closed for free amalgamation over self-sufficient substructures. Proof. δ(C) = δ(B) + δ(B′) − δ(A).

A B B′ C

= ⇒ C0 has a generalized Fraïssé limit M0.

EPPA C0 CF Summary

Hrushovski property of Hrushovski construction

EPPA (with joint embedding) is a stronger form of amalgamation.

A B B′ C

EPPA C0 CF Summary

Hrushovski property of Hrushovski construction

EPPA (with joint embedding) is a stronger form of amalgamation.

A B B′ C B B′

EPPA C0 CF Summary

Hrushovski property of Hrushovski construction

EPPA (with joint embedding) is a stronger form of amalgamation.

A B B′ C B B′

Question Does class C0 have EPPA (or a Hrushovski property) for partial automorphisms of self-sufficient substructures?

EPPA C0 CF Summary

Hrushovski property of Hrushovski construction

EPPA (with joint embedding) is a stronger form of amalgamation.

A B B′ C B B′

Question Does class C0 have EPPA (or a Hrushovski property) for partial automorphisms of self-sufficient substructures?

No!

Simple counter-example appears in disertation of Zaniar Ghadernezhad (2013).

EPPA C0 CF Summary

Hrushovski property of Hrushovski construction

EPPA (with joint embedding) is a stronger form of amalgamation.

A B B′ C B B′

Question Does class C0 have EPPA (or a Hrushovski property) for partial automorphisms of self-sufficient substructures?

No!

Simple counter-example appears in disertation of Zaniar Ghadernezhad (2013). In this talk we aim to understand the situation better.

EPPA C0 CF Summary

Orientations C0

Recall:

• Predimension of a graph G = (V, E) is δ(G) = 2|V| − |E|.
• Finite graph G is in C0 iff ∀H⊆Gδ(H) ≥ 0.

Lemma (By marriage theorem)

• G ∈ C0 iff it has 2-orientation (out-degrees at most 2).
• H is self sufficient in G iff G can be 2-oriented with no edge from H to G \ H.

EPPA C0 CF Summary

Orientations C0

Recall:

• Predimension of a graph G = (V, E) is δ(G) = 2|V| − |E|.
• Finite graph G is in C0 iff ∀H⊆Gδ(H) ≥ 0.

Lemma (By marriage theorem)

• G ∈ C0 iff it has 2-orientation (out-degrees at most 2).
• H is self sufficient in G iff G can be 2-oriented with no edge from H to G \ H.

EPPA C0 CF Summary

Orientations C0

Recall:

• Predimension of a graph G = (V, E) is δ(G) = 2|V| − |E|.
• Finite graph G is in C0 iff ∀H⊆Gδ(H) ≥ 0.

Lemma (By marriage theorem)

• G ∈ C0 iff it has 2-orientation (out-degrees at most 2).
• H is self sufficient in G iff G can be 2-oriented with no edge from H to G \ H.

EPPA C0 CF Summary

Orientations C0

Recall:

• Predimension of a graph G = (V, E) is δ(G) = 2|V| − |E|.
• Finite graph G is in C0 iff ∀H⊆Gδ(H) ≥ 0.

Lemma (By marriage theorem)

• G ∈ C0 iff it has 2-orientation (out-degrees at most 2).
• H is self sufficient in G iff G can be 2-oriented with no edge from H to G \ H.

Corollary C0 is, equivalently, created from class D0 of all finite 2-orientations by forgetting the

• rientation.

D0 is closed for free amalgamation over successor-closed substructures.

EPPA C0 CF Summary

Hrushovski classes has no Hrushovski property

We use: Theorem (Kechris, Rosendal 2007) Suppose K is an amalgamation class of finite structures with (generalised) Fraïssé limit

• M. Let Γ = Aut(M). Suppose K has EPPA then Γ is amenable.

EPPA C0 CF Summary

Hrushovski classes has no Hrushovski property

We use: Theorem (Kechris, Rosendal 2007) Suppose K is an amalgamation class of finite structures with (generalised) Fraïssé limit

• M. Let Γ = Aut(M). Suppose K has EPPA then Γ is amenable.

Recall: A topological group Γ is amenable if, whenever Y is a Γ-flow, then there is a Borel probability measure µ on Y which is invariant under the action of Γ.

EPPA C0 CF Summary

Hrushovski classes has no Hrushovski property

We use: Theorem (Kechris, Rosendal 2007) Suppose K is an amalgamation class of finite structures with (generalised) Fraïssé limit

• M. Let Γ = Aut(M). Suppose K has EPPA then Γ is amenable.

Recall: A topological group Γ is amenable if, whenever Y is a Γ-flow, then there is a Borel probability measure µ on Y which is invariant under the action of Γ. . . . and show: Theorem (Evans, H., Nešetˇ ril, 2019) Let M0 be a generalised Fraïssé limit of C0. Aut(M0) is not amenable. As a consequence of the two theorems C0 has no EPPA.

EPPA C0 CF Summary

Non-amenability of M0

Theorem (Evans, H., Nešetˇ ril, 2019) Let M be an 2-orientable graph and Γ is a topological group which acts continuously on

• M. Suppose there are adjacent vertices a, b in M such that the Γa-orbit containing b

and the Γb-orbit containing a are both infinite. Then Γ is not amenable. Proof.

EPPA C0 CF Summary

Non-amenability of M0

Theorem (Evans, H., Nešetˇ ril, 2019) Let M be an 2-orientable graph and Γ is a topological group which acts continuously on

• M. Suppose there are adjacent vertices a, b in M such that the Γa-orbit containing b

and the Γb-orbit containing a are both infinite. Then Γ is not amenable. Proof.

1 Suppose, for a contradiction, that µ is a Γ-invariant Borel probability measure on

the Γ-flow XM of 2-orientations. Let a, b be as in the statement.

2 Consider the open set Sab = {S ∈ XM : there is an edge oriented a → b}.

EPPA C0 CF Summary

Non-amenability of M0

Theorem (Evans, H., Nešetˇ ril, 2019) Let M be an 2-orientable graph and Γ is a topological group which acts continuously on

• M. Suppose there are adjacent vertices a, b in M such that the Γa-orbit containing b

and the Γb-orbit containing a are both infinite. Then Γ is not amenable. Proof.

1 Suppose, for a contradiction, that µ is a Γ-invariant Borel probability measure on

the Γ-flow XM of 2-orientations. Let a, b be as in the statement.

2 Consider the open set Sab = {S ∈ XM : there is an edge oriented a → b}. 3 As Sab ∪ Sba = XM we may assume that µ(Sab) = p = 0.

EPPA C0 CF Summary

Non-amenability of M0

Theorem (Evans, H., Nešetˇ ril, 2019) Let M be an 2-orientable graph and Γ is a topological group which acts continuously on

• M. Suppose there are adjacent vertices a, b in M such that the Γa-orbit containing b

and the Γb-orbit containing a are both infinite. Then Γ is not amenable. Proof.

1 Suppose, for a contradiction, that µ is a Γ-invariant Borel probability measure on

the Γ-flow XM of 2-orientations. Let a, b be as in the statement.

2 Consider the open set Sab = {S ∈ XM : there is an edge oriented a → b}. 3 As Sab ∪ Sba = XM we may assume that µ(Sab) = p = 0. 4 For r ∈ N, let b1, . . . , br be distinct elements of the Ga-orbit containing b. So

µ(Sabi ) = p for each i ≤ r. Let si be the characteristic function of Sabi .

EPPA C0 CF Summary

Non-amenability of M0

Theorem (Evans, H., Nešetˇ ril, 2019) Let M be an 2-orientable graph and Γ is a topological group which acts continuously on

• M. Suppose there are adjacent vertices a, b in M such that the Γa-orbit containing b

and the Γb-orbit containing a are both infinite. Then Γ is not amenable. Proof.

1 Suppose, for a contradiction, that µ is a Γ-invariant Borel probability measure on

the Γ-flow XM of 2-orientations. Let a, b be as in the statement.

2 Consider the open set Sab = {S ∈ XM : there is an edge oriented a → b}. 3 As Sab ∪ Sba = XM we may assume that µ(Sab) = p = 0. 4 For r ∈ N, let b1, . . . , br be distinct elements of the Ga-orbit containing b. So

µ(Sabi ) = p for each i ≤ r. Let si be the characteristic function of Sabi .

5 Then for every 2-orientation S ∈ XΓ we have i≤r si(S) ≤ 2.

EPPA C0 CF Summary

Non-amenability of M0

Theorem (Evans, H., Nešetˇ ril, 2019) Let M be an 2-orientable graph and Γ is a topological group which acts continuously on

• M. Suppose there are adjacent vertices a, b in M such that the Γa-orbit containing b

and the Γb-orbit containing a are both infinite. Then Γ is not amenable. Proof.

1 Suppose, for a contradiction, that µ is a Γ-invariant Borel probability measure on

the Γ-flow XM of 2-orientations. Let a, b be as in the statement.

2 Consider the open set Sab = {S ∈ XM : there is an edge oriented a → b}. 3 As Sab ∪ Sba = XM we may assume that µ(Sab) = p = 0. 4 For r ∈ N, let b1, . . . , br be distinct elements of the Ga-orbit containing b. So

µ(Sabi ) = p for each i ≤ r. Let si be the characteristic function of Sabi .

5 Then for every 2-orientation S ∈ XΓ we have i≤r si(S) ≤ 2.

Thus

• S∈XM
• i≤r

si(S) dµ(S) ≤ 2.

EPPA C0 CF Summary

Non-amenability of M0

Theorem (Evans, H., Nešetˇ ril, 2019) Let M be an 2-orientable graph and Γ is a topological group which acts continuously on

• M. Suppose there are adjacent vertices a, b in M such that the Γa-orbit containing b

and the Γb-orbit containing a are both infinite. Then Γ is not amenable. Proof.

1 Suppose, for a contradiction, that µ is a Γ-invariant Borel probability measure on

the Γ-flow XM of 2-orientations. Let a, b be as in the statement.

2 Consider the open set Sab = {S ∈ XM : there is an edge oriented a → b}. 3 As Sab ∪ Sba = XM we may assume that µ(Sab) = p = 0. 4 For r ∈ N, let b1, . . . , br be distinct elements of the Ga-orbit containing b. So

µ(Sabi ) = p for each i ≤ r. Let si be the characteristic function of Sabi .

5 Then for every 2-orientation S ∈ XΓ we have i≤r si(S) ≤ 2.

Thus

• S∈XM
• i≤r

si(S) dµ(S) ≤ 2. However

• S∈XM

si(S)dµ(S) = p.

EPPA C0 CF Summary

Non-amenability of M0

Theorem (Evans, H., Nešetˇ ril, 2019) Let M be an 2-orientable graph and Γ is a topological group which acts continuously on

• M. Suppose there are adjacent vertices a, b in M such that the Γa-orbit containing b

and the Γb-orbit containing a are both infinite. Then Γ is not amenable. Proof.

1 Suppose, for a contradiction, that µ is a Γ-invariant Borel probability measure on

the Γ-flow XM of 2-orientations. Let a, b be as in the statement.

2 Consider the open set Sab = {S ∈ XM : there is an edge oriented a → b}. 3 As Sab ∪ Sba = XM we may assume that µ(Sab) = p = 0. 4 For r ∈ N, let b1, . . . , br be distinct elements of the Ga-orbit containing b. So

µ(Sabi ) = p for each i ≤ r. Let si be the characteristic function of Sabi .

5 Then for every 2-orientation S ∈ XΓ we have i≤r si(S) ≤ 2.

Thus

• S∈XM
• i≤r

si(S) dµ(S) ≤ 2. However

• S∈XM

si(S)dµ(S) = p. therefore rp ≤ 2. As p = 0 and r is unbounded, this is a contradiction.

EPPA C0 CF Summary

Non-amenability of M0

Theorem (Evans, H., Nešetˇ ril, 2019) Let M be an 2-orientable graph and Γ is a topological group which acts continuously on

• M. Suppose there are adjacent vertices a, b in M such that the Γa-orbit containing b

and the Γb-orbit containing a are both infinite. Then Γ is not amenable. Proof.

1 Suppose, for a contradiction, that µ is a Γ-invariant Borel probability measure on

the Γ-flow XM of 2-orientations. Let a, b be as in the statement.

2 Consider the open set Sab = {S ∈ XM : there is an edge oriented a → b}. 3 As Sab ∪ Sba = XM we may assume that µ(Sab) = p = 0. 4 For r ∈ N, let b1, . . . , br be distinct elements of the Ga-orbit containing b. So

µ(Sabi ) = p for each i ≤ r. Let si be the characteristic function of Sabi .

5 Then for every 2-orientation S ∈ XΓ we have i≤r si(S) ≤ 2.

Thus

• S∈XM
• i≤r

si(S) dµ(S) ≤ 2. However

• S∈XM

si(S)dµ(S) = p. therefore rp ≤ 2. As p = 0 and r is unbounded, this is a contradiction. This is David Evans’ argument generalised by Todor Tsankov.

EPPA C0 CF Summary

Does D0 have EPPA?

Question Does class D0 (of all 2-orientations) have EPPA for partial automorphisms of successor-closed substructures?

EPPA C0 CF Summary

Does D0 have EPPA?

Question Does class D0 (of all 2-orientations) have EPPA for partial automorphisms of successor-closed substructures?

Yes!

EPPA C0 CF Summary

Does D0 have EPPA?

Question Does class D0 (of all 2-orientations) have EPPA for partial automorphisms of successor-closed substructures?

Yes!

We generalise notion of L-structures to represent self-sufficient substructures by functions.

EPPA C0 CF Summary

Does D0 have EPPA?

Question Does class D0 (of all 2-orientations) have EPPA for partial automorphisms of successor-closed substructures?

Yes!

We generalise notion of L-structures to represent self-sufficient substructures by functions. Let L be a language and A L-structure with domain A. Then function FA is from n-tuples of elements of A to subsets of A. FA : Aa → P(A).

EPPA C0 CF Summary

Does D0 have EPPA?

Question Does class D0 (of all 2-orientations) have EPPA for partial automorphisms of successor-closed substructures?

Yes!

We generalise notion of L-structures to represent self-sufficient substructures by functions. Let L be a language and A L-structure with domain A. Then function FA is from n-tuples of elements of A to subsets of A. FA : Aa → P(A). In this context D0 is a free amalgamation class of structures in language with single unary function F mapping every vertex to set of its successors in the 2-orientation.

EPPA C0 CF Summary

Does D0 have EPPA?

Question Does class D0 (of all 2-orientations) have EPPA for partial automorphisms of successor-closed substructures?

Yes!

We generalise notion of L-structures to represent self-sufficient substructures by functions. Let L be a language and A L-structure with domain A. Then function FA is from n-tuples of elements of A to subsets of A. FA : Aa → P(A). In this context D0 is a free amalgamation class of structures in language with single unary function F mapping every vertex to set of its successors in the 2-orientation.

F

EPPA C0 CF Summary

Does D0 have EPPA?

Question Does class D0 (of all 2-orientations) have EPPA for partial automorphisms of successor-closed substructures?

Yes!

We generalise notion of L-structures to represent self-sufficient substructures by functions. Let L be a language and A L-structure with domain A. Then function FA is from n-tuples of elements of A to subsets of A. FA : Aa → P(A). In this context D0 is a free amalgamation class of structures in language with single unary function F mapping every vertex to set of its successors in the 2-orientation.

F

EPPA C0 CF Summary

Does D0 have EPPA?

Question Does class D0 (of all 2-orientations) have EPPA for partial automorphisms of successor-closed substructures?

Yes!

We generalise notion of L-structures to represent self-sufficient substructures by functions. Let L be a language and A L-structure with domain A. Then function FA is from n-tuples of elements of A to subsets of A. FA : Aa → P(A). In this context D0 is a free amalgamation class of structures in language with single unary function F mapping every vertex to set of its successors in the 2-orientation. Theorem (Evans, H., Nešetˇ ril 2018+) Let L be a language consisting of relations and unary functions and K a free amalgamation class of L-structures. Then K has EPPA. This is a strengthening of earlier result of Hodkinson and Otto for relational languages. Open for functions of arbitrary arity.

EPPA C0 CF Summary

Does D0 have EPPA?

Question Does class D0 (of all 2-orientations) have EPPA for partial automorphisms of successor-closed substructures?

Yes!

We generalise notion of L-structures to represent self-sufficient substructures by functions. Let L be a language and A L-structure with domain A. Then function FA is from n-tuples of elements of A to subsets of A. FA : Aa → P(A). In this context D0 is a free amalgamation class of structures in language with single unary function F mapping every vertex to set of its successors in the 2-orientation. Theorem (Evans, H., Nešetˇ ril 2018+) Let L be a language consisting of relations and unary functions and K a free amalgamation class of L-structures. Then K has EPPA. This is a strengthening of earlier result of Hodkinson and Otto for relational languages. Open for functions of arbitrary arity. Question What is maximal amenable subgroup of Aut(M0)?

EPPA C0 CF Summary

The ω-categorical case

• F : R≥0 → R≥0

EPPA C0 CF Summary

The ω-categorical case

• F : R≥0 → R≥0
• C0 = {B : δ(A) ≥ 0 for all A ⊆ B}.

CF = {B : δ(A) ≥ F(|A|) for all A ⊆ B}.

EPPA C0 CF Summary

The ω-categorical case

• F : R≥0 → R≥0
• C0 = {B : δ(A) ≥ 0 for all A ⊆ B}.

CF = {B : δ(A) ≥ F(|A|) for all A ⊆ B}.

• A is self-sufficient in B iff δ(A) ≤ δ(B′) for all finite B′ with A ⊂ B′ ⊆ B.

A is d-closed in B iff δ(A) < δ(B′) for all finite B′ with A ⊂ B′ ⊆ B.

EPPA C0 CF Summary

The ω-categorical case

• F : R≥0 → R≥0
• C0 = {B : δ(A) ≥ 0 for all A ⊆ B}.

CF = {B : δ(A) ≥ F(|A|) for all A ⊆ B}.

• A is self-sufficient in B iff δ(A) ≤ δ(B′) for all finite B′ with A ⊂ B′ ⊆ B.

A is d-closed in B iff δ(A) < δ(B′) for all finite B′ with A ⊂ B′ ⊆ B. Lemma Put F(x) = ln(x). Then CF is a free amalgamation class over d-closed substructures.

EPPA C0 CF Summary

The ω-categorical case

• F : R≥0 → R≥0
• C0 = {B : δ(A) ≥ 0 for all A ⊆ B}.

CF = {B : δ(A) ≥ F(|A|) for all A ⊆ B}.

• A is self-sufficient in B iff δ(A) ≤ δ(B′) for all finite B′ with A ⊂ B′ ⊆ B.

A is d-closed in B iff δ(A) < δ(B′) for all finite B′ with A ⊂ B′ ⊆ B. Lemma Put F(x) = ln(x). Then CF is a free amalgamation class over d-closed substructures. Proof. A B B′ C A δ(G) |G| B B′ C Again there exists a generalized Fraïssé limit MF .

EPPA C0 CF Summary

The ω-categorical case

• F : R≥0 → R≥0
• C0 = {B : δ(A) ≥ 0 for all A ⊆ B}.

CF = {B : δ(A) ≥ F(|A|) for all A ⊆ B}.

• A is self-sufficient in B iff δ(A) ≤ δ(B′) for all finite B′ with A ⊂ B′ ⊆ B.

A is d-closed in B iff δ(A) < δ(B′) for all finite B′ with A ⊂ B′ ⊆ B. Lemma Put F(x) = ln(x). Then CF is a free amalgamation class over d-closed substructures. Proof. A B B′ C A δ(G) |G| B B′ C F(x) Again there exists a generalized Fraïssé limit MF .

EPPA C0 CF Summary

Successor-d-closure

Denote by rootsA(B) the set of all roots of A reachable from B ⊆ A. Lemma (H., Evans, Nešetˇ ril, 2019) Let B ⊆ A be 2-orientations. Then B is both d-closed and successor-closed in A iff B = {v : rootsA(v) ⊆ rootsA(B)}. Recall: B is d-closed in A iff δ(B) < δ(B′) for all B′ s.t. B ⊂ B′ ⊆ A.

EPPA C0 CF Summary

Successor-d-closure

Denote by rootsA(B) the set of all roots of A reachable from B ⊆ A. Lemma (H., Evans, Nešetˇ ril, 2019) Let B ⊆ A be 2-orientations. Then B is both d-closed and successor-closed in A iff B = {v : rootsA(v) ⊆ rootsA(B)}. Recall: B is d-closed in A iff δ(B) < δ(B′) for all B′ s.t. B ⊂ B′ ⊆ A. Proof.

• Given B ⊑s A, δ(B) is the number of roots of
• ut-degree 1 + twice number of roots of
• ut-degree 0.
• Extending B by all vertices v such that

rootsA(v) ⊆ rootsA(B) does not affect δ.

• Extending B by any other vertex increases δ.

EPPA C0 CF Summary

Free amalgamation class of 2-orientations of CF

1 Denote by DF the class of all 2-orientations. It is an amalgamation class for

successor-d-closed substructures.

EPPA C0 CF Summary

Free amalgamation class of 2-orientations of CF

1 Denote by DF the class of all 2-orientations. It is an amalgamation class for

successor-d-closed substructures.

2 DF is a free amalgamation class in language with functions:

Language L+ consists of a function symbol F of arity 1 and function symbols Fi of arity i for every i ≥ 1.

1 For every vertex v we put F(v) to be the set of all vertices v′ such that there

is edge v → v′.

EPPA C0 CF Summary

Free amalgamation class of 2-orientations of CF

1 Denote by DF the class of all 2-orientations. It is an amalgamation class for

successor-d-closed substructures.

2 DF is a free amalgamation class in language with functions:

F

Language L+ consists of a function symbol F of arity 1 and function symbols Fi of arity i for every i ≥ 1.

1 For every vertex v we put F(v) to be the set of all vertices v′ such that there

is edge v → v′.

EPPA C0 CF Summary

Free amalgamation class of 2-orientations of CF

1 Denote by DF the class of all 2-orientations. It is an amalgamation class for

successor-d-closed substructures.

2 DF is a free amalgamation class in language with functions:

F

Language L+ consists of a function symbol F of arity 1 and function symbols Fi of arity i for every i ≥ 1.

1 For every vertex v we put F(v) to be the set of all vertices v′ such that there

is edge v → v′.

EPPA C0 CF Summary

Free amalgamation class of 2-orientations of CF

1 Denote by DF the class of all 2-orientations. It is an amalgamation class for

successor-d-closed substructures.

2 DF is a free amalgamation class in language with functions:

F F2

Language L+ consists of a function symbol F of arity 1 and function symbols Fi of arity i for every i ≥ 1.

1 For every vertex v we put F(v) to be the set of all vertices v′ such that there

is edge v → v′.

2 For every n-tuple ¯

r of distinct root vertices we define Fn(¯ r) to be the set of all vertices v such that roots(v) is precisely ¯ r.

EPPA C0 CF Summary

Free amalgamation class of 2-orientations of CF

1 Denote by DF the class of all 2-orientations. It is an amalgamation class for

successor-d-closed substructures.

2 DF is a free amalgamation class in language with functions:

F F2

Language L+ consists of a function symbol F of arity 1 and function symbols Fi of arity i for every i ≥ 1.

1 For every vertex v we put F(v) to be the set of all vertices v′ such that there

is edge v → v′.

2 For every n-tuple ¯

r of distinct root vertices we define Fn(¯ r) to be the set of all vertices v such that roots(v) is precisely ¯ r.

3 Functions Fn(¯

r) = ∅ otherwise (set valued functions give free amalgamation good meaning).

EPPA C0 CF Summary

Free amalgamation class of 2-orientations of CF

1 Denote by DF the class of all 2-orientations. It is an amalgamation class for

successor-d-closed substructures.

2 DF is a free amalgamation class in language with functions:

F F2

Language L+ consists of a function symbol F of arity 1 and function symbols Fi of arity i for every i ≥ 1.

1 For every vertex v we put F(v) to be the set of all vertices v′ such that there

is edge v → v′.

2 For every n-tuple ¯

r of distinct root vertices we define Fn(¯ r) to be the set of all vertices v such that roots(v) is precisely ¯ r.

3 Functions Fn(¯

r) = ∅ otherwise (set valued functions give free amalgamation good meaning). Can we prove EPPA for a special case of free amalgamation class with non-unary functions?

EPPA C0 CF Summary

Symmetric version of L-structures with partial functions and permutation of the language

• Let L be a language with relation symbols and function symbols each with arity

denoted by a(R) and a(F ).

• We consider multiple-valued functions: for function symbol F ∈ L, L-structure A,

x ∈ A we put F A(x) ⊆ A.

EPPA C0 CF Summary

Symmetric version of L-structures with partial functions and permutation of the language

• Let L be a language with relation symbols and function symbols each with arity

denoted by a(R) and a(F ).

• We consider multiple-valued functions: for function symbol F ∈ L, L-structure A,

x ∈ A we put F A(x) ⊆ A.

• Let Γ

L be a permutation group on L which preserves types and arities of all

• symbols. We will say that Γ

L is a language equipped with a permutation group.

(This generalise Herwig’s notion of a permomorphism.)

EPPA C0 CF Summary

Symmetric version of L-structures with partial functions and permutation of the language

• Let L be a language with relation symbols and function symbols each with arity

denoted by a(R) and a(F ).

• We consider multiple-valued functions: for function symbol F ∈ L, L-structure A,

x ∈ A we put F A(x) ⊆ A.

• Let Γ

L be a permutation group on L which preserves types and arities of all

• symbols. We will say that Γ

L is a language equipped with a permutation group.

(This generalise Herwig’s notion of a permomorphism.) We consider Γ

L-structures which are essentially L-structures with the following

definition of homomorphism: Definition A homomorphism f : A → B is a pair f = (fL, fA) where fL ∈ Γ

L and fA is a mapping

A → B such that for every R ∈ LR and F ∈ LF we have: (a) (x1, x2, . . . , xa(R)) ∈ RA = ⇒ (fA(x1), fA(x2), . . . , fA(xa(R))) ∈ fL(R)B, and, (b) fA(FA(x1, c2, . . . , xa(F ))) ⊆ fL(F )B(fA(x1), fA(x2), . . . , fA(xa(F ))).

EPPA C0 CF Summary

Symmetric version of L-structures with partial functions and permutation of the language

• Let L be a language with relation symbols and function symbols each with arity

denoted by a(R) and a(F ).

• We consider multiple-valued functions: for function symbol F ∈ L, L-structure A,

x ∈ A we put F A(x) ⊆ A.

• Let Γ

L be a permutation group on L which preserves types and arities of all

• symbols. We will say that Γ

L is a language equipped with a permutation group.

(This generalise Herwig’s notion of a permomorphism.) We consider Γ

L-structures which are essentially L-structures with the following

definition of homomorphism: Definition A homomorphism f : A → B is a pair f = (fL, fA) where fL ∈ Γ

L and fA is a mapping

A → B such that for every R ∈ LR and F ∈ LF we have: (a) (x1, x2, . . . , xa(R)) ∈ RA = ⇒ (fA(x1), fA(x2), . . . , fA(xa(R))) ∈ fL(R)B, and, (b) fA(FA(x1, c2, . . . , xa(F ))) ⊆ fL(F )B(fA(x1), fA(x2), . . . , fA(xa(F ))). Notion of embedding, homomorphism-embedding, substructure generalise naturally to this category.

EPPA C0 CF Summary

EPPA for DF

Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) Let Γ

L be a language equipped with a permutation group consisting of relations and

unary functions and K a free amalgamation class of Γ

L-structures. Then K has EPPA.

Moreover for every A ∈ K the EPPA witness B ∈ K can be constructed such that every irreducible substructure of B is isomorphic to a substructure of A.

EPPA C0 CF Summary

EPPA for DF

Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) Let Γ

L be a language equipped with a permutation group consisting of relations and

unary functions and K a free amalgamation class of Γ

L-structures. Then K has EPPA.

Moreover for every A ∈ K the EPPA witness B ∈ K can be constructed such that every irreducible substructure of B is isomorphic to a substructure of A. Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) The class DF has EPPA for the successor-d-closed substructures

EPPA C0 CF Summary

EPPA for DF

Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) Let Γ

L be a language equipped with a permutation group consisting of relations and

unary functions and K a free amalgamation class of Γ

L-structures. Then K has EPPA.

Moreover for every A ∈ K the EPPA witness B ∈ K can be constructed such that every irreducible substructure of B is isomorphic to a substructure of A. Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) The class DF has EPPA for the successor-d-closed substructures Proof.

1 Given A ∈ DF ⊆ D0 construct B ∈ D0 such that every partial automorphism of

successor-closed substructures extend to an automorphism of B. (This is done by the easy construction shown earlier)

EPPA C0 CF Summary

EPPA for DF

Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) Let Γ

L be a language equipped with a permutation group consisting of relations and

unary functions and K a free amalgamation class of Γ

L-structures. Then K has EPPA.

Moreover for every A ∈ K the EPPA witness B ∈ K can be constructed such that every irreducible substructure of B is isomorphic to a substructure of A. Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) The class DF has EPPA for the successor-d-closed substructures Proof.

1 Given A ∈ DF ⊆ D0 construct B ∈ D0 such that every partial automorphism of

successor-closed substructures extend to an automorphism of B. (This is done by the easy construction shown earlier)

2 Define language L+ adding for every root vertex v ∈ B and every ordering ¯

r of |roots(v)| a new relational symbol Rv,¯

r of arity |roots(v)|.

EPPA C0 CF Summary

EPPA for DF

Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) Let Γ

L be a language equipped with a permutation group consisting of relations and

unary functions and K a free amalgamation class of Γ

L-structures. Then K has EPPA.

Moreover for every A ∈ K the EPPA witness B ∈ K can be constructed such that every irreducible substructure of B is isomorphic to a substructure of A. Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) The class DF has EPPA for the successor-d-closed substructures Proof.

1 Given A ∈ DF ⊆ D0 construct B ∈ D0 such that every partial automorphism of

successor-closed substructures extend to an automorphism of B. (This is done by the easy construction shown earlier)

2 Define language L+ adding for every root vertex v ∈ B and every ordering ¯

r of |roots(v)| a new relational symbol Rv,¯

r of arity |roots(v)|. 3 Define Γ L+ using the automorphism group of B.

EPPA C0 CF Summary

EPPA for DF

Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) Let Γ

L be a language equipped with a permutation group consisting of relations and

unary functions and K a free amalgamation class of Γ

L-structures. Then K has EPPA.

Moreover for every A ∈ K the EPPA witness B ∈ K can be constructed such that every irreducible substructure of B is isomorphic to a substructure of A. Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) The class DF has EPPA for the successor-d-closed substructures Proof.

1 Given A ∈ DF ⊆ D0 construct B ∈ D0 such that every partial automorphism of

successor-closed substructures extend to an automorphism of B. (This is done by the easy construction shown earlier)

2 Define language L+ adding for every root vertex v ∈ B and every ordering ¯

r of |roots(v)| a new relational symbol Rv,¯

r of arity |roots(v)|. 3 Define Γ L+ using the automorphism group of B. 4 Construct ΓL+ structure A+ by removing all non-root vertices and putting ¯

r ∈ Rv,¯

r A

for every non-root v ∈ A and ¯ r an ordering of roots(v).

EPPA C0 CF Summary

EPPA for DF

Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) Let Γ

L be a language equipped with a permutation group consisting of relations and

unary functions and K a free amalgamation class of Γ

L-structures. Then K has EPPA.

Moreover for every A ∈ K the EPPA witness B ∈ K can be constructed such that every irreducible substructure of B is isomorphic to a substructure of A. Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) The class DF has EPPA for the successor-d-closed substructures Proof.

1 Given A ∈ DF ⊆ D0 construct B ∈ D0 such that every partial automorphism of

successor-closed substructures extend to an automorphism of B. (This is done by the easy construction shown earlier)

2 Define language L+ adding for every root vertex v ∈ B and every ordering ¯

r of |roots(v)| a new relational symbol Rv,¯

r of arity |roots(v)|. 3 Define Γ L+ using the automorphism group of B. 4 Construct ΓL+ structure A+ by removing all non-root vertices and putting ¯

r ∈ Rv,¯

r A

for every non-root v ∈ A and ¯ r an ordering of roots(v).

5 A+ has only unary functions! Construct EPPA witness B+.

EPPA C0 CF Summary

EPPA for DF

Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) Let Γ

L be a language equipped with a permutation group consisting of relations and

unary functions and K a free amalgamation class of Γ

L-structures. Then K has EPPA.

Moreover for every A ∈ K the EPPA witness B ∈ K can be constructed such that every irreducible substructure of B is isomorphic to a substructure of A. Theorem (H., Koneˇ cný, Nešetˇ ril 2019+) The class DF has EPPA for the successor-d-closed substructures Proof.

1 Given A ∈ DF ⊆ D0 construct B ∈ D0 such that every partial automorphism of

successor-closed substructures extend to an automorphism of B. (This is done by the easy construction shown earlier)

2 Define language L+ adding for every root vertex v ∈ B and every ordering ¯

r of |roots(v)| a new relational symbol Rv,¯

r of arity |roots(v)|. 3 Define Γ L+ using the automorphism group of B. 4 Construct ΓL+ structure A+ by removing all non-root vertices and putting ¯

r ∈ Rv,¯

r A

for every non-root v ∈ A and ¯ r an ordering of roots(v).

5 A+ has only unary functions! Construct EPPA witness B+. 6 Construct B′ corresponding to B+ by adding the non-root vertices.

EPPA C0 CF Summary

Summary and open problems

• We can describe amenable subgroup of both Aut(M0) and Aut(MF ) by means of
• rientations and show EPPA. Our constructions generalise to some other variants
• f classes obtained by Hrushovski predimension constructions.

EPPA C0 CF Summary

Summary and open problems

• We can describe amenable subgroup of both Aut(M0) and Aut(MF ) by means of
• rientations and show EPPA. Our constructions generalise to some other variants
• f classes obtained by Hrushovski predimension constructions.
• For related class of acyclic orientations we know that such subgroup is maximal

amenable subgroup. Maximality for Aut(M0) and Aut(MF ) are open.

EPPA C0 CF Summary

Summary and open problems

• We can describe amenable subgroup of both Aut(M0) and Aut(MF ) by means of
• rientations and show EPPA. Our constructions generalise to some other variants
• f classes obtained by Hrushovski predimension constructions.
• For related class of acyclic orientations we know that such subgroup is maximal

amenable subgroup. Maximality for Aut(M0) and Aut(MF ) are open.

• Techniques developed here can be applied to solve several other cases, such as

EPPA for two-graphs, antipodal metric spaces, n-partite tournaments, semigeneric tournaments, . . .

EPPA C0 CF Summary

Summary and open problems

• We can describe amenable subgroup of both Aut(M0) and Aut(MF ) by means of
• rientations and show EPPA. Our constructions generalise to some other variants
• f classes obtained by Hrushovski predimension constructions.
• For related class of acyclic orientations we know that such subgroup is maximal

amenable subgroup. Maximality for Aut(M0) and Aut(MF ) are open.

• Techniques developed here can be applied to solve several other cases, such as

EPPA for two-graphs, antipodal metric spaces, n-partite tournaments, semigeneric tournaments, . . .

• We can also give simpler proofs for EPPA for metric spaces and other strong

amalgamation classes

EPPA C0 CF Summary

Summary and open problems

• We can describe amenable subgroup of both Aut(M0) and Aut(MF ) by means of
• rientations and show EPPA. Our constructions generalise to some other variants
• f classes obtained by Hrushovski predimension constructions.
• For related class of acyclic orientations we know that such subgroup is maximal

amenable subgroup. Maximality for Aut(M0) and Aut(MF ) are open.

• Techniques developed here can be applied to solve several other cases, such as

EPPA for two-graphs, antipodal metric spaces, n-partite tournaments, semigeneric tournaments, . . .

• We can also give simpler proofs for EPPA for metric spaces and other strong

amalgamation classes

• ΓL-languages and functions combine well with conditions given by Herwig-Lascar

theorem leading to so far strongest sufficient structural condition for EPPA.

EPPA C0 CF Summary

Summary and open problems

• We can describe amenable subgroup of both Aut(M0) and Aut(MF ) by means of
• rientations and show EPPA. Our constructions generalise to some other variants
• f classes obtained by Hrushovski predimension constructions.
• For related class of acyclic orientations we know that such subgroup is maximal

amenable subgroup. Maximality for Aut(M0) and Aut(MF ) are open.

• Techniques developed here can be applied to solve several other cases, such as

EPPA for two-graphs, antipodal metric spaces, n-partite tournaments, semigeneric tournaments, . . .

• We can also give simpler proofs for EPPA for metric spaces and other strong

amalgamation classes

• ΓL-languages and functions combine well with conditions given by Herwig-Lascar

theorem leading to so far strongest sufficient structural condition for EPPA. However many open questions remain

• does the class of all finite tournaments have EPPA?

This is a classical open question in the are asked by Herwig and Lascar

EPPA C0 CF Summary

Summary and open problems

• We can describe amenable subgroup of both Aut(M0) and Aut(MF ) by means of
• rientations and show EPPA. Our constructions generalise to some other variants
• f classes obtained by Hrushovski predimension constructions.
• For related class of acyclic orientations we know that such subgroup is maximal

amenable subgroup. Maximality for Aut(M0) and Aut(MF ) are open.

• Techniques developed here can be applied to solve several other cases, such as

EPPA for two-graphs, antipodal metric spaces, n-partite tournaments, semigeneric tournaments, . . .

• We can also give simpler proofs for EPPA for metric spaces and other strong

amalgamation classes

• ΓL-languages and functions combine well with conditions given by Herwig-Lascar

theorem leading to so far strongest sufficient structural condition for EPPA. However many open questions remain

• does the class of all finite tournaments have EPPA?

This is a classical open question in the are asked by Herwig and Lascar

• Does the class of all finite partial Steiner systems (or structures with non-unary

functions in general) have EPPA?

EPPA C0 CF Summary

Summary and open problems

• We can describe amenable subgroup of both Aut(M0) and Aut(MF ) by means of
• rientations and show EPPA. Our constructions generalise to some other variants
• f classes obtained by Hrushovski predimension constructions.
• For related class of acyclic orientations we know that such subgroup is maximal

amenable subgroup. Maximality for Aut(M0) and Aut(MF ) are open.

• Techniques developed here can be applied to solve several other cases, such as

EPPA for two-graphs, antipodal metric spaces, n-partite tournaments, semigeneric tournaments, . . .

• We can also give simpler proofs for EPPA for metric spaces and other strong

amalgamation classes

• ΓL-languages and functions combine well with conditions given by Herwig-Lascar

theorem leading to so far strongest sufficient structural condition for EPPA. However many open questions remain

• does the class of all finite tournaments have EPPA?

This is a classical open question in the are asked by Herwig and Lascar

• Does the class of all finite partial Steiner systems (or structures with non-unary

functions in general) have EPPA?

• Does the class of all finite structures with one quaternary relation defining two

equivalence classes on pairs have EPPA?

EPPA C0 CF Summary

Summary and open problems

• We can describe amenable subgroup of both Aut(M0) and Aut(MF ) by means of
• rientations and show EPPA. Our constructions generalise to some other variants
• f classes obtained by Hrushovski predimension constructions.
• For related class of acyclic orientations we know that such subgroup is maximal

amenable subgroup. Maximality for Aut(M0) and Aut(MF ) are open.

• Techniques developed here can be applied to solve several other cases, such as

EPPA for two-graphs, antipodal metric spaces, n-partite tournaments, semigeneric tournaments, . . .

• We can also give simpler proofs for EPPA for metric spaces and other strong

amalgamation classes

• ΓL-languages and functions combine well with conditions given by Herwig-Lascar

theorem leading to so far strongest sufficient structural condition for EPPA. However many open questions remain

• does the class of all finite tournaments have EPPA?

This is a classical open question in the are asked by Herwig and Lascar

• Does the class of all finite partial Steiner systems (or structures with non-unary

functions in general) have EPPA?

• Does the class of all finite structures with one quaternary relation defining two

equivalence classes on pairs have EPPA?

• Can we obtain general structural condition for the existence of EPPA?

EPPA C0 CF Summary

Original class

EPPA C0 CF Summary

Original class

EPPA C0 CF Summary

Precompact Original class

EPPA C0 CF Summary

Precompact Expansion property Original class

EPPA C0 CF Summary

Precompact Expansion property Original class Extremely amenable KPT

EPPA C0 CF Summary

Precompact amenable Expansion property Original class Extremely amenable KPT

EPPA C0 CF Summary

Precompact amenable Expansion property Original class Extremely amenable Max EA Max A

EPPA C0 CF Summary

Thank you for the attention

• D. Evans, J.H., J. Nešetˇ

ril: Automorphism groups and Ramsey properties of sparse graphs. Proceedings of the London Mathematical Society 119 (2) (2019), 515-546.

• J.H., J. Nešetˇ

ril: All those EPPA classes (A strengthening of the Herwig-Lascar theorem). arXiv:1902.03855 (2019).

• D. Evans, J. H., J. Nešetˇ

ril: Ramsey properties and extending partial automorphisms for classes of finite structures. Submitted (arXiv:1705.02379).

• I. Hodkinson, M. Otto: Finite conformal hypergraph covers and Gaifman cliques in finite structures, Bulletin of Symbolic Logic 3 (9)

(2013), 387–405.

• J.H., J. Nešetˇ

ril: All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms). Accepted to Advances in Mathematics (arXiv:1606.07979), 2019.

• J.H., M. Koneˇ

cný, J. Nešetˇ ril: A combinatorial proof of the extension property for partial isometries. Commentationes Mathematicae Universitatis Carolinae 60 (1) 2019, 39–47

• A. Aranda, D. Bradley-Williams, J. H., M. Karamanlis, M. Kompatscher, M. Koneˇ

cný, M. Pawliuk: Ramsey expansions of metrically homogeneous graphs. Submitted (arXiv:1706.00295), 57 pages.

• J.H., M. Koneˇ

cný, J. Nešetˇ ril: Conant’s generalised metric spaces are Ramsey. To appear in Contributions to Discrete Mathematics (arXiv:1710.04690), 20 pages.

• D. Evans, J.H., M. Koneˇ

cný, J. Nešetˇ ril: EPPA for two-graphs and antipodal metric spaces. arXiv:1812.11157 (2018).

• J. Hubiˇ

cka, C. Jahel, M. Koneˇ cný, M. Sabok: Extension property for partial automorphisms of the n-partite and semi-generic

• tournaments. To appear.