All those Ramsey classes Ramsey classes with closures and forbidden - PowerPoint PPT Presentation

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints All those Ramsey classes Ramsey classes with closures and forbidden homomorphisms Jan Hubiˇ cka Computer Science Institute of Charles University Charles University Prague Joint work with Jaroslav Nešetˇ ril Logic Colloquium 2016, Leeds

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Ramsey Theorem Theorem (Ramsey Theorem, 1930) → ( n ) p ∀ n , p , k ≥ 1 ∃ N : N − k .

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Ramsey Theorem Theorem (Ramsey Theorem, 1930) → ( n ) p ∀ n , p , k ≥ 1 ∃ N : N − k . � { 1 , 2 ,..., N } � → ( n ) p N − k : For every partition of into k classes p � X � (colors) there exists X ⊆ { 1 , 2 , . . . , N } , | X | = n such that p belongs to single partition (it is monochromatic)

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Ramsey Theorem Theorem (Ramsey Theorem, 1930) → ( n ) p ∀ n , p , k ≥ 1 ∃ N : N − k . � { 1 , 2 ,..., N } � → ( n ) p N − k : For every partition of into k classes p � X � (colors) there exists X ⊆ { 1 , 2 , . . . , N } , | X | = n such that p belongs to single partition (it is monochromatic) For p = 2, n = 3, k = 2 put N = 6

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Many aspects of Ramsey theorem Ramsey theorem

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Many aspects of Ramsey theorem Ramsey theorem Logic

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Many aspects of Ramsey theorem Ramsey theorem Logic Combinatorics

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Many aspects of Ramsey theorem Ramsey theorem Logic Combinatorics Model Theory

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Many aspects of Ramsey theorem Ramsey theorem Logic Topological Combinatorics Model Theory dynamics

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints 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 ∀ A , B ∈− Rel ( L ) ∃ C ∈− Rel ( L ) : C − → → 2 .

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints 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 ∀ A , B ∈− Rel ( L ) ∃ C ∈− Rel ( L ) : C − → → 2 . Theorem (Ramsey Theorem, 1930) → ( n ) p ∀ n , p , k ≥ 1 ∃ N : N − k .

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints 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 ∀ A , B ∈− Rel ( L ) ∃ C ∈− Rel ( L ) : C − → → 2 . � B � is the set of all substructures of B isomorphic to A . A � C � � C � there exists � → ( B ) A C − 2 : For every 2-colouring of B ∈ such A B � � � B that is monochromatic. A

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints 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 ∀ A , B ∈− Rel ( L ) ∃ C ∈− Rel ( L ) : C − → → 2 . � B � is the set of all substructures of B isomorphic to A . A � C � � C � there exists � → ( B ) A C − 2 : For every 2-colouring of B ∈ such A B � � � B that is monochromatic. A C A B

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints 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 ∀ A , B ∈− Rel ( L ) ∃ C ∈− Rel ( L ) : C − → → 2 . � B � is the set of all substructures of B isomorphic to A . A � C � � C � there exists � → ( B ) A C − 2 : For every 2-colouring of B ∈ such A B � � � B that is monochromatic. A C A B

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Order is necessary B A

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints 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.

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Structural extensions Ramsey theorem for finite relational structures

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Structural extensions Ramsey theorem for finite relational structures Structures with functions (Models)

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Structural extensions Ramsey theorem for finite relational structures Structures Structures with with functions axioms (Models)

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Structural extensions Ramsey theorem for finite relational structures Structures Structures Infinite with with structures functions axioms (Models)

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Structural extensions Ramsey theorem for finite relational structures Categories Structures Structures Infinite with with structures functions axioms (Models)

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Ramsey theorem for finite models Let L be a language with both relations and functions. Assume that L contains binary relation ≤ . Denote by − − → Mod ( L ) the class of all finite L -structures where ≤ is a linear order. Theorem (H.-Nešetˇ ril, 2016) → ( B ) A ∀ A , B ∈− Mod ( L ) ∃ C ∈− Mod ( L ) : C − − → − → 2 .

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Ramsey classes Definition → ( B ) A A class C of finite L -structures is Ramsey iff ∀ A , B ∈C ∃ C ∈C : C − 2 .

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints 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.

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints 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 , − − → Rel ( L ) is a Ramsey class.

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints 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 , − − → Rel ( L ) 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.

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints 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 , − − → Rel ( L ) 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. Example (Models — H.-Nešetˇ ril, 2016) For every language L , − − → Mod ( L ) is a Ramsey class.

Ramsey → Structural Ramsey Ramsey classes Ramsey classes with constraints Ramsey classes are amalgamation classes Definition (Amalgamation property of class K ) B A C B ′