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Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Ramsey properties of Hrushovski construction Jan Hubi cka Computer Science Institute of Charles University Charles University Prague

  1. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Nešetˇ ril’s Classification Programme, 2005 Classification Programme = ⇒ Ramsey classes amalgamation classes ⇑ ⇓ expansions of homogeneous ⇐ = homogeneous structures ⇓ ⇑ ⇓ extremely amenable groups = ⇒ universal minimal flows Kechris, Pestov, Todorˇ cevi` c: Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups (2005) Definition Let L ′ be language containing language L . A expansion (or lift) of L -structure A is L ′ -structure A ′ on the same vertex set such that all relations/functions in L ∩ L ′ are identical.

  2. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Nešetˇ ril’s Classification Programme, 2005 Classification Programme = ⇒ Ramsey classes amalgamation classes ⇑ ⇓ expansions of homogeneous ⇐ = homogeneous structures ⇓ ⇑ ⇓ extremely amenable groups = ⇒ universal minimal flows Kechris, Pestov, Todorˇ cevi` c: Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups (2005) Definition Let L ′ be language containing language L . A expansion (or lift) of L -structure A is L ′ -structure A ′ on the same vertex set such that all relations/functions in L ∩ L ′ are identical. Theorem (Nešetˇ ril, 1989) All homogeneous graphs have Ramsey expansion.

  3. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Nešetˇ ril’s Classification Programme, 2005 Classification Programme amalgamation classes Example 1 The class of finite graphs G is an amalgamation class

  4. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Nešetˇ ril’s Classification Programme, 2005 Classification Programme amalgamation classes ⇓ homogeneous structures Example 1 The class of finite graphs G is an amalgamation class 2 Fraïssé limit of G is the Rado graph R

  5. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Nešetˇ ril’s Classification Programme, 2005 Classification Programme amalgamation classes ⇓ expansions of homogeneous ⇐ = homogeneous structures Example 1 The class of finite graphs G is an amalgamation class 2 Fraïssé limit of G is the Rado graph R 3 The expansion R ′ of R adds generic linear order

  6. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Nešetˇ ril’s Classification Programme, 2005 Classification Programme Ramsey classes amalgamation classes ⇑ ⇓ expansions of homogeneous ⇐ = homogeneous structures Example 1 The class of finite graphs G is an amalgamation class 2 Fraïssé limit of G is the Rado graph R 3 The expansion R ′ of R adds generic linear order 4 Age ( R ′ ) (the class of linearly ordered finite graphs) is Ramsey

  7. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Nešetˇ ril’s Classification Programme, 2005 Classification Programme Ramsey classes = ⇒ amalgamation classes ⇑ ⇓ expansions of homogeneous ⇐ = homogeneous structures Example 1 The class of finite graphs G is an amalgamation class 2 Fraïssé limit of G is the Rado graph R 3 The expansion R ′ of R adds generic linear order 4 Age ( R ′ ) (the class of linearly ordered finite graphs) is Ramsey

  8. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Milliken tree theorem: C-relations Ramsey’s theorem: rationals Product arguments

  9. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Milliken tree theorem: C-relations Permutations Equivalences Ramsey’s theorem: rationals Product arguments

  10. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Boolean algebras Semilattices Milliken tree theorem: C-relations Permutations Interval graphs Cyclic orders Equivalences Ramsey’s theorem: rationals Product arguments Interpretations

  11. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Boolean algebras Semilattices Milliken tree theorem: C-relations Permutations Interval graphs Unary functions Cyclic orders Equivalences Ramsey’s theorem: rationals Product arguments Interpretations Adding unary functions

  12. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Boolean algebras Semilattices Milliken tree theorem: C-relations Permutations Interval graphs Unary functions Free amalgamation classes Cyclic orders Equivalences − → Ramsey’s theorem: rationals Rel( L ) Product arguments Interpretations Adding unary functions Partite construction

  13. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Graham Rotschild Theorem: Parametric words Boolean algebras Semilattices Partial Steiner systems Milliken tree theorem: C-relations Metric spaces Permutations Interval graphs Unary functions Free amalgamation classes Cyclic orders Equivalences − → Ramsey’s theorem: rationals Rel( L ) Product arguments Interpretations Adding unary functions Partite construction

  14. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Lelek fans Graham Rotschild Theorem: Parametric words Boolean algebras Boolean algebras with ideals Semilattices Partial Steiner systems Milliken tree theorem: C-relations Metric spaces Permutations Interval graphs Structures with unary functions Unary functions Free amalgamation classes Cyclic orders Partial orders Equivalences Acyclic graphs Line graphs − → Ramsey’s theorem: rationals Rel( L ) Product arguments Interpretations Adding unary functions Partite construction

  15. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Does every amalgamation class have a Ramsey expansion? Question (Bodirsky, Nešetˇ ril, Nguyen van Thé, Pinsker, Tsankov 2010) Does every amalgamation class have a Ramsey expansion?

  16. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Does every amalgamation class have a Ramsey expansion? Question (Bodirsky, Nešetˇ ril, Nguyen van Thé, Pinsker, Tsankov 2010) Does every amalgamation class have a Ramsey expansion? Yes: extend language by infinitely many unary relations; assign every vertex to unique relation.

  17. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Does every amalgamation class have a Ramsey expansion? Question (Bodirsky, Nešetˇ ril, Nguyen van Thé, Pinsker, Tsankov 2010) Does every amalgamation class have a Ramsey expansion? Yes: extend language by infinitely many unary relations; assign every vertex to unique relation. Definition (Nguyen van Thé) Let K be class of L -structures and K ′ be class of expansions of K . • K ′ is precompact wrt K if for every A ∈ K there are only finitely many expansions of A in K ′ .

  18. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Does every amalgamation class have a Ramsey expansion? Question (Bodirsky, Nešetˇ ril, Nguyen van Thé, Pinsker, Tsankov 2010) Does every amalgamation class have a Ramsey expansion? Yes: extend language by infinitely many unary relations; assign every vertex to unique relation. Definition (Nguyen van Thé) Let K be class of L -structures and K ′ be class of expansions of K . • K ′ is precompact wrt K if for every A ∈ K there are only finitely many expansions of A in K ′ . • K ′ has expansion property if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ .

  19. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Does every amalgamation class have a Ramsey expansion? Question (Bodirsky, Nešetˇ ril, Nguyen van Thé, Pinsker, Tsankov 2010) Does every amalgamation class have a Ramsey expansion? Yes: extend language by infinitely many unary relations; assign every vertex to unique relation. Definition (Nguyen van Thé) Let K be class of L -structures and K ′ be class of expansions of K . • K ′ is precompact wrt K if for every A ∈ K there are only finitely many expansions of A in K ′ . • K ′ has expansion property if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Theorem (Kechris, Pestov, Todorˇ cevi` c 2005, Nguyen van Thé 2012) For every amalgamation class K there exists, up to bi-definability, at most one Ramsey class K ′ of precompact expansions of K with expansion property.

  20. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Existence of precompact expansions Question Does every amalgamation class have a precompact Ramsey expansion? No: Consider Z seen as a metric space.

  21. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Existence of precompact expansions Question Does every amalgamation class have a precompact Ramsey expansion? No: Consider Z seen as a metric space. Better question (Nguyen Van Thé) Does every ω -categorical structure have a precompact Ramsey expansion?

  22. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Existence of precompact expansions Question Does every amalgamation class have a precompact Ramsey expansion? No: Consider Z seen as a metric space. Better question (Nguyen Van Thé) Does every ω -categorical structure have a precompact Ramsey expansion?

  23. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Existence of precompact expansions Question Does every amalgamation class have a precompact Ramsey expansion? No: Consider Z seen as a metric space. Better question (Nguyen Van Thé) Does every ω -categorical structure have a precompact Ramsey expansion?

  24. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Lelek fans Graham Rotschild Theorem: Parametric words Boolean algebras Boolean algebras with ideals Semilattices Partial Steiner systems Milliken tree theorem: C-relations Metric spaces Permutations Interval graphs Structures with unary functions Unary functions Free amalgamation classes Cyclic orders Partial orders Equivalences Acyclic graphs Line graphs − → Ramsey’s theorem: rationals Rel( L ) Product arguments Interpretations Adding unary functions Partite construction

  25. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Lelek fans Graham Rotschild Theorem: Parametric words Boolean algebras Boolean algebras with ideals Semilattices Partial Steiner systems Models (Structures with functions) Milliken tree theorem: C-relations Metric spaces Permutations Interval graphs Structures with unary functions Unary functions Free amalgamation classes Cyclic orders Partial orders Equivalences Acyclic graphs Line graphs − → Ramsey’s theorem: rationals Rel( L ) Locally finite subclass Product arguments Interpretations Adding unary functions Partite construction

  26. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Lelek fans Graham Rotschild Theorem: Parametric words Boolean algebras Boolean algebras with ideals Semilattices Partial Steiner systems S -metric spaces Models (Structures with functions) Milliken tree theorem: C-relations Metric spaces Permutations Cherlin Shelah Shi classes Metrically homogeneous graphs Interval graphs Structures with unary functions Unary functions Free amalgamation classes Cyclic orders Partial orders Equivalences Acyclic graphs Line graphs − → Ramsey’s theorem: rationals Rel( L ) Locally finite subclass Product arguments Interpretations Adding unary functions Partite construction

  27. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Lelek fans Graham Rotschild Theorem: Parametric words Boolean algebras Boolean algebras with ideals Semilattices Partial Steiner systems Models (Structures with functions) Milliken tree theorem: C-relations S -metric spaces Metric spaces Permutations Cherlin Shelah Shi classes Metrically homogeneous graphs Interval graphs Structures with unary functions Structures with unary functions Free amalgamation classes Cyclic orders Partial orders Equivalences Acyclic graphs Line graphs − → Ramsey’s theorem: rationals Rel( L ) Locally finite subclass Product arguments Interpretations Adding unary functions Partite construction

  28. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Existence of precompact expansions Theorem (Evans, 2015+) There is a countable, ω -categorical structure M F no precompact Ramsey expansion. In this talk we explore properties of this example.

  29. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Three variants of David’s example • C 0 : The easy example • C 1 : The kindergarten example • C F : The actual counter-example

  30. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski 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 .

  31. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski 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.

  32. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski 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, G ≤ s H , iff ∀ G ⊆ G ′ ⊆ H δ ( G ) ≤ δ ( G ′ ) .

  33. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski 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, G ≤ s H , iff ∀ G ⊆ G ′ ⊆ H δ ( G ) ≤ δ ( G ′ ) . Lemma C 0 is closed for free amalgamation over self-sufficient C B substructures. A B ′ Proof. δ ( C ) = δ ( B ) + δ ( B ′ ) − δ ( A ) .

  34. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski class C 0 as a reduct • 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 ≤ s G iff G can be 2 -oriented with no edge from H to G \ H .

  35. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski class C 0 as a reduct • 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 ≤ s G iff G can be 2 -oriented with no edge from H to G \ H .

  36. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski class C 0 as a reduct • 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 ≤ s G iff G can be 2 -oriented with no edge from H to G \ H .

  37. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski class C 0 as a reduct • 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 ≤ s G iff G can be 2 -oriented with no edge from H to G \ H . Corollary C 0 is a reduct of the class of all finite 2 -orientations D 0 . D 0 is closed for free amalgamation over successor-closed substructures.

  38. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Ramsey expansions of C 0 and orientations Theorem (Kechris, Pestov, Todorˇ cevi` c, 2005) Let F be a Fraïssé limit, then the following are equivalent. • Automorphism group of F is extremely amenable; • Age ( F ) has the Ramsey property.

  39. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Ramsey expansions of C 0 and orientations Theorem (Kechris, Pestov, Todorˇ cevi` c, 2005) Let F be a Fraïssé limit, then the following are equivalent. • Automorphism group of F is extremely amenable; • Age ( F ) has the Ramsey property. Denote by M 0 the generalised Fraïssé limit of C 0 . Theorem (Evans 2015) If M + 0 is a Ramsey expansion of M 0 , then Aut ( M + 0 ) fixes a 2 -orientation.

  40. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Ramsey expansions of C 0 and orientations Theorem (Kechris, Pestov, Todorˇ cevi` c, 2005) Let F be a Fraïssé limit, then the following are equivalent. • Automorphism group of F is extremely amenable; • Age ( F ) has the Ramsey property. Denote by M 0 the generalised Fraïssé limit of C 0 . Theorem (Evans 2015) If M + 0 is a Ramsey expansion of M 0 , then Aut ( M + 0 ) fixes a 2 -orientation. Proof. • Consider G acting on the space X ( M 0 ) of 2-orientations of M 0 (a G -flow). • As Aut ( M + 0 ) is extremely amenable, there is some S ∈ X ( M 0 ) which is fixed by Aut ( M + 0 ) . • Aut ( M + 0 ) is a subgroup of Aut ( S ) .

  41. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F No precompact Ramsey expansions of C 0 Theorem (Evans 2016) There is no precompact Ramsey expansion of ( C 0 ; ≤ s ) .

  42. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F No precompact Ramsey expansions of C 0 Theorem (Evans 2016) There is no precompact Ramsey expansion of ( C 0 ; ≤ s ) . • Let ( C + 0 , ⊑ ) be a Ramsey expansion of ( C 0 , ≤ s ) , then every A ∈ C 0 has infinitely many expansions in ( C + 0 ; ⊑ ) .

  43. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F No precompact Ramsey expansions of C 0 Theorem (Evans 2016) There is no precompact Ramsey expansion of ( C 0 ; ≤ s ) . • Let ( C + 0 , ⊑ ) be a Ramsey expansion of ( C 0 , ≤ s ) , then every A ∈ C 0 has infinitely many expansions in ( C + 0 ; ⊑ ) . • Given two 2-orientations A ⊆ B , we write A ⊑ s B if there is no edge from A to B \ A . • ⊑ is coarser than ⊑ s for 2-orientation fixed by ( C + 0 , ⊑ ) .

  44. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F No precompact Ramsey expansions of C 0 Theorem (Evans 2016) There is no precompact Ramsey expansion of ( C 0 ; ≤ s ) . • Let ( C + 0 , ⊑ ) be a Ramsey expansion of ( C 0 , ≤ s ) , then every A ∈ C 0 has infinitely many expansions in ( C + 0 ; ⊑ ) . • Given two 2-orientations A ⊆ B , we write A ⊑ s B if there is no edge from A to B \ A . • ⊑ is coarser than ⊑ s for 2-orientation fixed by ( C + 0 , ⊑ ) .

  45. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F No precompact Ramsey expansions of C 0 Theorem (Evans 2016) There is no precompact Ramsey expansion of ( C 0 ; ≤ s ) . • Let ( C + 0 , ⊑ ) be a Ramsey expansion of ( C 0 , ≤ s ) , then every A ∈ C 0 has infinitely many expansions in ( C + 0 ; ⊑ ) . • Given two 2-orientations A ⊆ B , we write A ⊑ s B if there is no edge from A to B \ A . • ⊑ is coarser than ⊑ s for 2-orientation fixed by ( C + 0 , ⊑ ) . Proof. • Every vertex v ∈ M + 0 has out-degree at most 2, but infinite in-degree. • Oriented path v 1 → v 2 → v 2 . . . v n always extendeds by a vertex v 0 to v 0 → v 1 → v 2 → v 2 . . . v n .

  46. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey Denote by D ≺ 0 the class of all finite ordered 2-orientations. Theorem (H., Evans, Nešetˇ ril, 2015+) D ≺ 0 is a Ramsey class.

  47. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey Denote by D ≺ 0 the class of all finite ordered 2-orientations. Theorem (H., Evans, Nešetˇ ril, 2015+) D ≺ 0 is a Ramsey class. Proof. → ( | B | ) | A | • Given A , B ∈ D ≺ 0 put N − 2 . • Extend language by unary predicates R 1 , R 2 , . . . R N . • Given | B | tuple � b = ( b 1 , b 2 , . . . b | B | ) , denote by B � b expansion of B where i -th vertex is in relation R b i . � n � • P 0 is a disjoint union of B � v , v ∈ . | B | • Put u ∼ v if successor-closure of u is isomorphic to v . → ( B ) A • C = P 0 / ∼ . C − 2 .

  48. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey B A

  49. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey B A 1 2 3 4 5

  50. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey B A 1 2 3 4 5

  51. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey B A 1 2 3 4 5

  52. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Optimality of Ramsey expansion Question: (Tsankov) Is ( D ≺ 0 ; ⊑ s ) any better than the trivial Ramsey expansion?

  53. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Optimality of Ramsey expansion Question: (Tsankov) Is ( D ≺ 0 ; ⊑ s ) any better than the trivial Ramsey expansion? Theorem (H., Evans, Nešetˇ ril, 2016+) There exists G 0 ⊂ D ≺ 0 such that • ( G 0 ; ⊑ s ) is strong expansion of ( C 0 ; ≤ s ) , • ( G 0 ; ⊑ s ) is Ramsey classes, • N G 0 , the group of automorphisms of Fraïssé limit of ( G 0 ; ⊑ s ) is maximal amongst extremely amenable subgroups of Aut ( M 0 ) . • Class of all self-sufficient substructures of G 0 has an Expansion Property with respect to C 0 and thus give a minimal Aut ( M 0 ) flow.

  54. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ .

  55. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Denote by ( D 1 ; ⊑ s ) the class of all finite acyclic orientations. Denote by ( C 1 ; ⊑ s ) unoriented reduct of ( D 1 ; ⊑ s ) .

  56. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Denote by ( D 1 ; ⊑ s ) the class of all finite acyclic orientations. Denote by ( C 1 ; ⊑ s ) unoriented reduct of ( D 1 ; ⊑ s ) . Theorem For every A + ∈ D 1 there exists B ∈ C 1 such that every expansion B + ∈ D 1 contains A + as a self-sufficient substructure.

  57. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Denote by ( D 1 ; ⊑ s ) the class of all finite acyclic orientations. Denote by ( C 1 ; ⊑ s ) unoriented reduct of ( D 1 ; ⊑ s ) . Theorem For every A + ∈ D 1 there exists B ∈ C 1 such that every expansion B + ∈ D 1 contains A + as a self-sufficient substructure. Proof by induction on | A + | . v

  58. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Denote by ( D 1 ; ⊑ s ) the class of all finite acyclic orientations. Denote by ( C 1 ; ⊑ s ) unoriented reduct of ( D 1 ; ⊑ s ) . Theorem For every A + ∈ D 1 there exists B ∈ C 1 such that every expansion B + ∈ D 1 contains A + as a self-sufficient substructure. Proof by induction on | A + | . v • Every A ∈ D 1 has vertex v of in-degree 0. • A 0 = A \ { v } .

  59. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Denote by ( D 1 ; ⊑ s ) the class of all finite acyclic orientations. Denote by ( C 1 ; ⊑ s ) unoriented reduct of ( D 1 ; ⊑ s ) . Theorem For every A + ∈ D 1 there exists B ∈ C 1 such that every expansion B + ∈ D 1 contains A + as a self-sufficient substructure. Proof by induction on | A + | . v • Every A ∈ D 1 has vertex v of in-degree 0. • A 0 = A \ { v } . • Construct B 0 by induction hypothesis. • Extend every copy of A 0 in B 0 to A by 5 copies of v .

  60. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Extension property of non-precompact expansion Definition Suppose A ∈ D 1 we put A ∈ E 1 iff: 1 If l ( a ) ≺ l ( b ) . 2 If l ( a ) = l ( b ) then order is defined lexicographically by descending chains of their successors l ( a ) denote the level of vertex a .

  61. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Extension property of non-precompact expansion Definition Suppose A ∈ D 1 we put A ∈ E 1 iff: 1 If l ( a ) ≺ l ( b ) . 2 If l ( a ) = l ( b ) then order is defined lexicographically by descending chains of their successors l ( a ) denote the level of vertex a . Theorem (H., Evans, Nešetˇ ril, 2016+) For every A + ∈ E 1 there exists B ∈ C 1 such that every expansion B + ∈ E 1 contains A as self-sufficient substructure.

  62. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Theorem (H., Evans, Nešetˇ ril, 2016+) For every A + ∈ E 1 there exists B ∈ C 1 such that every expansion B + ∈ E 1 contains A as self-sufficient substructure. Proof.

  63. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Theorem (H., Evans, Nešetˇ ril, 2016+) For every A + ∈ E 1 there exists B ∈ C 1 such that every expansion B + ∈ E 1 contains A as self-sufficient substructure. Proof.

  64. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Theorem (H., Evans, Nešetˇ ril, 2016+) For every A + ∈ E 1 there exists B ∈ C 1 such that every expansion B + ∈ E 1 contains A as self-sufficient substructure. Proof.

  65. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Theorem (H., Evans, Nešetˇ ril, 2016+) For every A + ∈ E 1 there exists B ∈ C 1 such that every expansion B + ∈ E 1 contains A as self-sufficient substructure. Proof.

  66. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has no Hrushovski property Given strong class ( C ; ≤ ) , a strong partial automorphism of A ∈ C is an isomorphism f : D → E for some D , E ≤ A .

  67. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has no Hrushovski property Given strong class ( C ; ≤ ) , a strong partial automorphism of A ∈ C is an isomorphism f : D → E for some D , E ≤ A . Definition ( C , ≤ ) has the extension property for strong partial automorphisms (EPPA) if ∀ A ∈C ∃ B ∈C , A ≤ B such that every strong partial automorphism of A extends to an automorphism of B .

  68. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has no Hrushovski property Given strong class ( C ; ≤ ) , a strong partial automorphism of A ∈ C is an isomorphism f : D → E for some D , E ≤ A . Definition ( C , ≤ ) has the extension property for strong partial automorphisms (EPPA) if ∀ A ∈C ∃ B ∈C , A ≤ B such that every strong partial automorphism of A extends to an automorphism of B . Theorem (Evans, 2016, easier argument by Tsankov) Aut ( M 0 ) is not amenable and thus ( C 0 ; ≤ s ) has no EPPA. Explicit example given by Zaniar Ghadernezhad.

  69. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has no Hrushovski property Given strong class ( C ; ≤ ) , a strong partial automorphism of A ∈ C is an isomorphism f : D → E for some D , E ≤ A . Definition ( C , ≤ ) has the extension property for strong partial automorphisms (EPPA) if ∀ A ∈C ∃ B ∈C , A ≤ B such that every strong partial automorphism of A extends to an automorphism of B . Theorem (Evans, 2016, easier argument by Tsankov) Aut ( M 0 ) is not amenable and thus ( C 0 ; ≤ s ) has no EPPA. Explicit example given by Zaniar Ghadernezhad. Theorem (H., Evans, Nešetˇ ril, 2017+) The class of all finite 2 -orientations has EPPA.

  70. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has Hrushovski expansion Theorem (H., Evans, Nešetˇ ril, 2017+) Let D be a class of finite 2 -orientations closed for free amalgamation over successor-closed substructures. Then D has EPPA.

  71. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has Hrushovski expansion Theorem (H., Evans, Nešetˇ ril, 2017+) Let D be a class of finite 2 -orientations closed for free amalgamation over successor-closed substructures. Then D has EPPA. Proof. • Given A ∈ D construct B 0 ∈ D as follows: 1 Vertices of B 0 are pairs ( v , f ) where v ∈ A and f ∈ Sym ( B ) . 2 ( v , f ) → ( v ′ , f ′ ) iff f = f ′ and f ( v ) → f ( v ′ ) is edge of A .

  72. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has Hrushovski expansion Theorem (H., Evans, Nešetˇ ril, 2017+) Let D be a class of finite 2 -orientations closed for free amalgamation over successor-closed substructures. Then D has EPPA. Proof. • Given A ∈ D construct B 0 ∈ D as follows: 1 Vertices of B 0 are pairs ( v , f ) where v ∈ A and f ∈ Sym ( B ) . 2 ( v , f ) → ( v ′ , f ′ ) iff f = f ′ and f ( v ) → f ( v ′ ) is edge of A . • Put ( v , f ) ∼ ( v , f ′ ) iff there is isomorphism of successor-closures α of ( v , f ) and ( v , f ′ ) such that α ( u , h ) = ( h , h ′ ) . • B = B 0 / ∼ .

  73. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has Hrushovski expansion Theorem (H., Evans, Nešetˇ ril, 2017+) Let D be a class of finite 2 -orientations closed for free amalgamation over successor-closed substructures. Then D has EPPA. Proof. • Given A ∈ D construct B 0 ∈ D as follows: 1 Vertices of B 0 are pairs ( v , f ) where v ∈ A and f ∈ Sym ( B ) . 2 ( v , f ) → ( v ′ , f ′ ) iff f = f ′ and f ( v ) → f ( v ′ ) is edge of A . • Put ( v , f ) ∼ ( v , f ′ ) iff there is isomorphism of successor-closures α of ( v , f ) and ( v , f ′ ) such that α ( u , h ) = ( h , h ′ ) . • B = B 0 / ∼ . Along with Herwig-Lascar theorem this also shows EPPA for unary Cherlin-Shelah-Shi classes and more.

  74. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Summary 1 ( C 1 ; ≤ s ) (reducts of acyclic 2-orientations) ¬ Ramsey, ¬ EPPA, AP 2 ( D 1 ; ⊑ s ) (acyclic 2-orientations) EP wrt C 1 , ¬ Ramsey, EPPA, Minimal flow, AP

  75. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Summary 1 ( C 1 ; ≤ s ) (reducts of acyclic 2-orientations) ¬ Ramsey, ¬ EPPA, AP 2 ( D 1 ; ⊑ s ) (acyclic 2-orientations) EP wrt C 1 , ¬ Ramsey, EPPA, Minimal flow, AP 3 ( D ≺ 1 ; ⊑ s ) (ordered acyclic 2-orientations) ¬ EP wrt C 1 nor D 1 , Ramsey, ¬ EPPA, ¬ Minimal flow, AP

  76. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Summary 1 ( C 1 ; ≤ s ) (reducts of acyclic 2-orientations) ¬ Ramsey, ¬ EPPA, AP 2 ( D 1 ; ⊑ s ) (acyclic 2-orientations) EP wrt C 1 , ¬ Ramsey, EPPA, Minimal flow, AP 3 ( D ≺ 1 ; ⊑ s ) (ordered acyclic 2-orientations) ¬ EP wrt C 1 nor D 1 , Ramsey, ¬ EPPA, ¬ Minimal flow, AP 4 ( E 1 ; ⊑ s ) (admisively ordered acyclic 2-orientations) EP wrt C 1 and D 1 Ramsey, ¬ EPPA, ¬ Minimal flow, AP

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