Ramsey Classes by Partite Construction II Honza Hubiˇ cka Mathematics and Statistics University of Calgary Calgary Institute of Computer Science Charles University Prague Joint work with Jaroslav Nešetˇ ril Permutation Groups and Transformation Semigroups 2015 J. Hubiˇ cka Ramsey Classes by Partite Construction II
Ramsey classes We consider relational structures in language L without function symbols. Definition A class C (of finite relational structures) is Ramsey iff → ( B ) A ∀ A , B ∈C ∃ C ∈C : C − 2 . J. Hubiˇ cka Ramsey Classes by Partite Construction II
Ramsey classes We consider relational structures in language L without function symbols. Definition A class C (of finite relational structures) is Ramsey iff → ( B ) A ∀ A , B ∈C ∃ C ∈C : C − 2 . � B � is set of all substructures of B isomorphic to A . A � C � � C � there exists � → ( B ) A C − 2 : For every 2-coloring of B ∈ A B � � � B such that is monochromatic. A J. Hubiˇ cka Ramsey Classes by Partite Construction II
Ramsey classes We consider relational structures in language L without function symbols. Definition A class C (of finite relational structures) is Ramsey iff → ( B ) A ∀ A , B ∈C ∃ C ∈C : C − 2 . � B � is set of all substructures of B isomorphic to A . A � C � � C � there exists � → ( B ) A C − 2 : For every 2-coloring of B ∈ A B � � � B such that is monochromatic. A C A B J. Hubiˇ cka Ramsey Classes by Partite Construction II
Ramsey classes We consider relational structures in language L without function symbols. Definition A class C (of finite relational structures) is Ramsey iff → ( B ) A ∀ A , B ∈C ∃ C ∈C : C − 2 . � B � is set of all substructures of B isomorphic to A . A � C � � C � there exists � → ( B ) A C − 2 : For every 2-coloring of B ∈ A B � � � B such that is monochromatic. A C A B J. Hubiˇ cka Ramsey Classes by Partite Construction II
Nešetˇ ril-Rödl Theorem A structure A is called complete (or irreducible) if every pair of distinct vertices belong to a relation of A . Forb E ( E ) is a class of all finite structures A such that there is no embedding from E ∈ E to A . Theorem (Nešetˇ ril-Rödl Theorem, 1977) Let L be a finite relational language. Let E be a set of complete ordered L-structures. The then class Forb E ( E ) is a Ramsey class. J. Hubiˇ cka Ramsey Classes by Partite Construction II
Nešetˇ ril-Rödl Theorem A structure A is called complete (or irreducible) if every pair of distinct vertices belong to a relation of A . Forb E ( E ) is a class of all finite structures A such that there is no embedding from E ∈ E to A . Theorem (Nešetˇ ril-Rödl Theorem, 1977) Let L be a finite relational language. Let E be a set of complete ordered L-structures. The then class Forb E ( E ) is a Ramsey class. Proof by partite construction. J. Hubiˇ cka Ramsey Classes by Partite Construction II
Unary closures = relations with out-degree 1 Unary closure description C is a set of pairs ( R U , R B ) where R U is unary relation and R B is binary relation. We say that structure A is C -closed if for every pair ( R U , R B ) the B -outdegree of every vertex of A that is in U is 1. Theorem (H., Nešetˇ ril, 2015) Let E be a family of complete ordered structures and U an unary closure description. Then the class of all C -closed structures in Forb E ( E ) has Ramsey lift. J. Hubiˇ cka Ramsey Classes by Partite Construction II
Unary closures = relations with out-degree 1 Unary closure description C is a set of pairs ( R U , R B ) where R U is unary relation and R B is binary relation. We say that structure A is C -closed if for every pair ( R U , R B ) the B -outdegree of every vertex of A that is in U is 1. Theorem (H., Nešetˇ ril, 2015) Let E be a family of complete ordered structures and U an unary closure description. Then the class of all C -closed structures in Forb E ( E ) has Ramsey lift. All Cherlin Shelah Shi classes with unary closure can be described this way! J. Hubiˇ cka Ramsey Classes by Partite Construction II
J. Hubiˇ cka Ramsey Classes by Partite Construction II
J. Hubiˇ cka Ramsey Classes by Partite Construction II
J. Hubiˇ cka Ramsey Classes by Partite Construction II
J. Hubiˇ cka Ramsey Classes by Partite Construction II
J. Hubiˇ cka Ramsey Classes by Partite Construction II
J. Hubiˇ cka Ramsey Classes by Partite Construction II
Map of Ramsey Classes boolean algebras Unary CSS classes interval graphs metric spaces permutations acyclic graphs cyclic orders partial orders unions of complete graphs K n -free graphs graphs linear orders restricted free J. Hubiˇ cka Ramsey Classes by Partite Construction II
Further applications Known Cherlin-Shelah-Shi classes: bouquets bowties extended by path known examples without unary closure J. Hubiˇ cka Ramsey Classes by Partite Construction II
Further applications Known Cherlin-Shelah-Shi classes: bouquets bowties extended by path known examples without unary closure n -ary functions (structures with a function symbol ( A , f ) where f : A n → A ) Consider ( A , f ) as a relational structure with ( n + 1 ) -ary relation where every n -tuple has a closure vertex Because the algebraic closure is not locally finite Fraïssé limit is not ω -categorical J. Hubiˇ cka Ramsey Classes by Partite Construction II
Further applications Known Cherlin-Shelah-Shi classes: bouquets bowties extended by path known examples without unary closure n -ary functions (structures with a function symbol ( A , f ) where f : A n → A ) Consider ( A , f ) as a relational structure with ( n + 1 ) -ary relation where every n -tuple has a closure vertex Because the algebraic closure is not locally finite Fraïssé limit is not ω -categorical In some cases algebraic closure is introduced as a scaffolding and does not appear in the final Ramsey class: Structures with infinitely many equivalence classes J. Hubiˇ cka Ramsey Classes by Partite Construction II
Further applications Known Cherlin-Shelah-Shi classes: bouquets bowties extended by path known examples without unary closure n -ary functions (structures with a function symbol ( A , f ) where f : A n → A ) Consider ( A , f ) as a relational structure with ( n + 1 ) -ary relation where every n -tuple has a closure vertex Because the algebraic closure is not locally finite Fraïssé limit is not ω -categorical In some cases algebraic closure is introduced as a scaffolding and does not appear in the final Ramsey class: Structures with infinitely many equivalence classes QQ J. Hubiˇ cka Ramsey Classes by Partite Construction II
Structures with forbidden homomorphisms Let F be a family of relational structures. We denote by Forb H ( F ) the class of all finite structures A such that there is no F ∈ F having a homomorphism F → A . J. Hubiˇ cka Ramsey Classes by Partite Construction II
Structures with forbidden homomorphisms Let F be a family of relational structures. We denote by Forb H ( F ) the class of all finite structures A such that there is no F ∈ F having a homomorphism F → A . Theorem (Cherlin,Shelah,Shi 1998) For every finite family F of finite connected relational structures there is an ω -categorical structure that is universal for Forb H ( F ) . J. Hubiˇ cka Ramsey Classes by Partite Construction II
Structures with forbidden homomorphisms Let F be a family of relational structures. We denote by Forb H ( F ) the class of all finite structures A such that there is no F ∈ F having a homomorphism F → A . Theorem (Cherlin,Shelah,Shi 1998) For every finite family F of finite connected relational structures there is an ω -categorical structure that is universal for Forb H ( F ) . Every ω -categorical structure can be lifted to homogeneous. Explicit homogenization is given by H. and Nešetˇ ril (2009). J. Hubiˇ cka Ramsey Classes by Partite Construction II
Structures with forbidden homomorphisms Let F be a family of relational structures. We denote by Forb H ( F ) the class of all finite structures A such that there is no F ∈ F having a homomorphism F → A . Theorem (Cherlin,Shelah,Shi 1998) For every finite family F of finite connected relational structures there is an ω -categorical structure that is universal for Forb H ( F ) . Every ω -categorical structure can be lifted to homogeneous. Explicit homogenization is given by H. and Nešetˇ ril (2009). Theorem (Nešetˇ ril, 2010) For every finite family F of finite connected relational structures there is a Ramsey lift of Forb H ( F ) . J. Hubiˇ cka Ramsey Classes by Partite Construction II
Explicit homogenization of Forb H ( C 5 ) Basic concept: Amalgamation of two structures in Forb H ( F ) fails iff the free amalgam contains a homomorphic copy of structure F ∈ F . Use extra relations to prevent such amalgams F J. Hubiˇ cka Ramsey Classes by Partite Construction II
Explicit homogenization of Forb H ( C 5 ) Basic concept: Amalgamation of two structures in Forb H ( F ) fails iff the free amalgam contains a homomorphic copy of structure F ∈ F . Use extra relations to prevent such amalgams + F J. Hubiˇ cka Ramsey Classes by Partite Construction II
Recommend
More recommend
