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

EPPA C 0 C F 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 C 0 C F Summary Definition (Extension property for partial automorphisms) A class C of finite L -structures has extension property for partial automorphisms (EPPA or Hrushovski property) iff for every A ∈ C

EPPA C 0 C F Summary Definition (Extension property for partial automorphisms) A class C of finite L -structures has extension property for partial automorphisms (EPPA or Hrushovski property) iff for every A ∈ C A

EPPA C 0 C F Summary Definition (Extension property for partial automorphisms) A class C of finite L -structures has extension property for partial automorphisms (EPPA or Hrushovski property) iff for every A ∈ C there exists EPPA witness B ∈ C containing A B A

EPPA C 0 C F Summary Definition (Extension property for partial automorphisms) A class C of finite L -structures has extension property for partial automorphisms (EPPA or 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. B A

EPPA C 0 C F Summary Definition (Extension property for partial automorphisms) A class C of finite L -structures has extension property for partial automorphisms (EPPA or 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. B A

EPPA C 0 C F Summary Definition (Extension property for partial automorphisms) A class C of finite L -structures has extension property for partial automorphisms (EPPA or 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 C 0 C F Summary Definition (Extension property for partial automorphisms) A class C of finite L -structures has extension property for partial automorphisms (EPPA or 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 C 0 C F Summary Hrushovski (predimension) construction • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . Example δ ( K 1 ) = 2 δ ( K 2 ) = 4 − 1 = 3 δ ( K 3 ) = 6 − 3 = 3 δ ( K 4 ) = 8 − 6 = 2 δ ( K 5 ) = 10 − 10 = 0 δ ( K 6 ) = 12 − 30 = − 18 .

EPPA C 0 C F Summary Hrushovski (predimension) construction • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . Example δ ( K 1 ) = 2 δ ( K 2 ) = 4 − 1 = 3 δ ( K 3 ) = 6 − 3 = 3 δ ( K 4 ) = 8 − 6 = 2 δ ( K 5 ) = 10 − 10 = 0 δ ( K 6 ) = 12 − 30 = − 18 . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0.

EPPA C 0 C F Summary Hrushovski (predimension) construction • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . Example δ ( K 1 ) = 2 δ ( K 2 ) = 4 − 1 = 3 δ ( K 3 ) = 6 − 3 = 3 δ ( K 4 ) = 8 − 6 = 2 δ ( K 5 ) = 10 − 10 = 0 δ ( K 6 ) = 12 − 30 = − 18 . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. • G ⊆ H is self-sufficient iff ∀ G ⊆ G ′ ⊆ H δ ( G ) ≤ δ ( G ′ ) .

EPPA C 0 C F Summary Hrushovski (predimension) construction • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . Example δ ( K 1 ) = 2 δ ( K 2 ) = 4 − 1 = 3 δ ( K 3 ) = 6 − 3 = 3 δ ( K 4 ) = 8 − 6 = 2 δ ( K 5 ) = 10 − 10 = 0 δ ( K 6 ) = 12 − 30 = − 18 . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. • G ⊆ H is self-sufficient iff ∀ G ⊆ G ′ ⊆ H δ ( G ) ≤ δ ( G ′ ) . Definition (Amalgamation property of class K ) B A C B ′

EPPA C 0 C F Summary Hrushovski (predimension) construction • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . Example δ ( K 1 ) = 2 δ ( K 2 ) = 4 − 1 = 3 δ ( K 3 ) = 6 − 3 = 3 δ ( K 4 ) = 8 − 6 = 2 δ ( K 5 ) = 10 − 10 = 0 δ ( K 6 ) = 12 − 30 = − 18 . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. • G ⊆ H is self-sufficient iff ∀ G ⊆ G ′ ⊆ H δ ( G ) ≤ δ ( G ′ ) . Definition (Amalgamation property of class K ) B A C B ′ Lemma C B C 0 is closed for free amalgamation over A self-sufficient substructures. B ′ Proof. δ ( C ) = δ ( B ) + δ ( B ′ ) − δ ( A ) .

EPPA C 0 C F Summary Hrushovski (predimension) construction • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . Example δ ( K 1 ) = 2 δ ( K 2 ) = 4 − 1 = 3 δ ( K 3 ) = 6 − 3 = 3 δ ( K 4 ) = 8 − 6 = 2 δ ( K 5 ) = 10 − 10 = 0 δ ( K 6 ) = 12 − 30 = − 18 . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. • G ⊆ H is self-sufficient iff ∀ G ⊆ G ′ ⊆ H δ ( G ) ≤ δ ( G ′ ) . Definition (Amalgamation property of class K ) B A C B ′ Lemma C B C 0 is closed for free amalgamation over A self-sufficient substructures. B ′ Proof. = ⇒ C 0 has a generalized Fraïssé limit M 0 . δ ( C ) = δ ( B ) + δ ( B ′ ) − δ ( A ) .

EPPA C 0 C F Summary Hrushovski property of Hrushovski construction EPPA (with joint embedding) is a stronger form of amalgamation. B A C B ′

EPPA C 0 C F Summary Hrushovski property of Hrushovski construction EPPA (with joint embedding) is a stronger form of amalgamation. B B A C B ′ B ′

EPPA C 0 C F Summary Hrushovski property of Hrushovski construction EPPA (with joint embedding) is a stronger form of amalgamation. B B A C B ′ B ′ Question Does class C 0 have EPPA (or a Hrushovski property) for partial automorphisms of self-sufficient substructures?

EPPA C 0 C F Summary Hrushovski property of Hrushovski construction EPPA (with joint embedding) is a stronger form of amalgamation. B B A C B ′ B ′ Question Does class C 0 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 C 0 C F Summary Hrushovski property of Hrushovski construction EPPA (with joint embedding) is a stronger form of amalgamation. B B A C B ′ B ′ Question Does class C 0 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 C 0 C F Summary Orientations C 0 Recall: • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. Lemma (By marriage theorem) • G ∈ C 0 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 C 0 C F Summary Orientations C 0 Recall: • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. Lemma (By marriage theorem) • G ∈ C 0 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 C 0 C F Summary Orientations C 0 Recall: • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. Lemma (By marriage theorem) • G ∈ C 0 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 C 0 C F Summary Orientations C 0 Recall: • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. Lemma (By marriage theorem) • G ∈ C 0 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 C 0 is, equivalently, created from class D 0 of all finite 2 -orientations by forgetting the orientation. D 0 is closed for free amalgamation over successor-closed substructures.