Ramsey theorems for classes of structures with functions and - PowerPoint PPT Presentation

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Ramsey theorems for classes of structures with functions and relations Jan Hubiˇ cka Department of Applied Mathematics Charles University Prague Joint work with David Evans, Matˇ ej Koneˇ cný and Jaroslav Nešetˇ ril Model Theory and Combinatorics 2018

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Ramsey theorem for finite relational structures Let L be a purely relational language with binary relation ≤ . Denote by − → Rel ( L ) the class of all finite L -structures where ≤ is a linear order. Theorem (Nešetˇ ril-Rödl, 1977; Abramson-Harrington, 1978) → ( B ) A Rel ( L ) : C − ∀ A , B ∈− Rel ( L ) ∃ C ∈− → → 2 .

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Ramsey theorem for finite relational structures Let L be a purely relational language with binary relation ≤ . Denote by − → Rel ( L ) the class of all finite L -structures where ≤ is a linear order. Theorem (Nešetˇ ril-Rödl, 1977; Abramson-Harrington, 1978) → ( B ) A Rel ( L ) : C − ∀ A , B ∈− Rel ( L ) ∃ C ∈− → → 2 . Theorem (Ramsey Theorem, 1930) → ( n ) p ∀ n , p , k ≥ 1 ∃ N : N − k .

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Ramsey theorem for finite relational structures Let L be a purely relational language with binary relation ≤ . Denote by − → Rel ( L ) the class of all finite L -structures where ≤ is a linear order. Theorem (Nešetˇ ril-Rödl, 1977; Abramson-Harrington, 1978) → ( B ) A Rel ( L ) : C − ∀ A , B ∈− Rel ( L ) ∃ C ∈− → → 2 . � B � is the set of all substructures of B isomorphic to A . A � C � � C � � � � there exists � → ( B ) A B C − 2 : For every 2-colouring of B ∈ such that is A B A monochromatic.

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Ramsey theorem for finite relational structures Let L be a purely relational language with binary relation ≤ . Denote by − → Rel ( L ) the class of all finite L -structures where ≤ is a linear order. Theorem (Nešetˇ ril-Rödl, 1977; Abramson-Harrington, 1978) → ( B ) A Rel ( L ) : C − ∀ A , B ∈− Rel ( L ) ∃ C ∈− → → 2 . � B � is the set of all substructures of B isomorphic to A . A � C � � C � � � � there exists � → ( B ) A B C − 2 : For every 2-colouring of B ∈ such that is A B A monochromatic. C A B

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Ramsey theorem for finite relational structures Let L be a purely relational language with binary relation ≤ . Denote by − → Rel ( L ) the class of all finite L -structures where ≤ is a linear order. Theorem (Nešetˇ ril-Rödl, 1977; Abramson-Harrington, 1978) → ( B ) A Rel ( L ) : C − ∀ A , B ∈− Rel ( L ) ∃ C ∈− → → 2 . � B � is the set of all substructures of B isomorphic to A . A � C � � C � � � � there exists � → ( B ) A B C − 2 : For every 2-colouring of B ∈ such that is A B A monochromatic. C A B

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Order is necessary B A

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Order is necessary B A Vertices of C can be linearly ordered and edges coloured accordingly: • If edge is goes forward in linear order it is red • blue otherwise.

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Ramsey classes Definition → ( B ) A A class C of finite L -structures is Ramsey iff ∀ A , B ∈C ∃ C ∈C : C − 2 . Example (Linear orders — Ramsey Theorem, 1930) The class of all finite linear orders is a Ramsey class. Example (Structures — Nešetˇ ril-Rödl, 76; Abramson-Harrington, 78) For every relational language L , class of all finite ordered L -structures is a Ramsey class. Example (Partial orders — Nešetˇ ril-Rödl, 84; Paoli-Trotter-Walker, 85) The class of all finite partial orders with linear extension is Ramsey.

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Ramsey classes are amalgamation classes Definition (Amalgamation) B A C B ′ Nešetˇ ril, 80’s: Under mild assumptions Ramsey classes have amalgamation property. C B A A

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Fraïssé limits Definition (Amalgamation class) A class K of finite relational structures is called an amalgamation class if the following conditions hold: 1 K is hereditary (closed under substructures). 2 K is closed under isomorphisms. 3 K has only countably many mutually non-isomorphic structures. 4 K has the amalgamation property B A C B ′ A structure A is homogeneous if every isomorphism of two induced finite substructures of A can be extended to an automorphism of A . Age ( U ) is the class of all finite structures isomorphic to a substructure of U . Theorem (Fraïssé) A class K of finite structures is the age of a countable homogeneous structure G if and only if K is an amalgamation class.

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Nešetˇ ril’s Classification Programme, 2005 Classification Programme = Ramsey classes ⇒ amalgamation classes ⇑ ⇓ expansions of homogeneous = homogeneous structures ⇐

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems 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)

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems 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 of L -structure A is L ′ -structure A ′ on the same vertex set such that all relations/functions in L ∩ L ′ are identical.

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems 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 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.

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Milliken tree theorem: C-relations Ramsey’s theorem: rationals Product arguments

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Milliken tree theorem: C-relations Permutations Equivalences Ramsey’s theorem: rationals Product arguments

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems 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

Structural Ramsey Free amalgamation classes Amalgamation with closures General amalgamation Open problems 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