ramsey classes with algebraical closure and forbidden ......examples of ramsey classes example the...

Ramsey classes with algebraical closure andforbidden homomorphisms

Honza Hubicka

Institute of Computer ScienceCharles University

Prague

Mathematics and StatisticsUniversity of Calgary

Calgary

Joint work with Jaroslav Nešetril

Algebraic and Model Theoretical Methods in ConstraintSatisfaction

J. Hubicka Ramsey classes with algebraical closure and forbidden homomorphisms

Ramsey classes

We consider relational structures in language L without functionsymbols.

DefinitionA class C (of finite relational structures) is Ramsey iff

∀A,B∈C∃C∈C : C −→ (B)A2 .

)is set of all substructures of B isomorphic to A.

C −→ (B)A2 : For every 2-coloring of

)there exists B ∈

)such that

)is monochromatic.

Ramsey classes

∀A,B∈C∃C∈C : C −→ (B)A2 .(B

)there exists B ∈

)such that

)is monochromatic.

Ramsey classes

∀A,B∈C∃C∈C : C −→ (B)A2 .(B

)there exists B ∈

)such that

)is monochromatic.

Ramsey classes

∀A,B∈C∃C∈C : C −→ (B)A2 .(B

)there exists B ∈

)such that

)is monochromatic.

Examples of Ramsey classes

ExampleThe class of all finite linear orders is Ramsey.

Example (Non-example)The class of all directed graphs is not Ramsey.

Every C can be linearly ordered. If edge is goes forward inlinear order it is red, blue otherwise.

Nešetril-Rödl Theorem

A structure A is called complete (or irreducible) if every pair ofdistinct vertices belong to a relation of A.

Theorem (Nešetril-Rödl Theorem, 1977)Let L be a finite relational language involving unaryrelations U1,U2, . . . ,UN .Let E be a set of complete ordered L-structures.

Let C be the class of all ordered of L-structuresA ∈ ForbE(E) where the admissible ordering ≤A satisfies

x < y whenever (x) ∈ U iA and (y) ∈ U j

A, i < j .

Then the class C together with admissible orderings is aRamsey class.

ForbE(E) is a class of all finite structures A such that there is noembedding from E ∈ E to A.

Theorem (Nešetril-Rödl Theorem, 1977)Let L be a finite relational language involving unaryrelations U1,U2, . . . ,UN .Let E be a set of complete ordered L-structures.Let C be the class of all ordered of L-structuresA ∈ ForbE(E) where the admissible ordering ≤A satisfies

A, i < j .

Theorem (Nešetril-Rödl Theorem, 1977)Let L be a finite relational language involving unaryrelations U1,U2, . . . ,UN .Let E be a set of complete ordered L-structures.Let C be the class of all ordered of L-structuresA ∈ ForbE(E) where the admissible ordering ≤A satisfies

A, i < j .

Examples of Ramsey lifts

ExampleGraphs with order are Ramsey.

ExampleAcyclic graphs with linear extension are Ramsey.

ExampleBipartite graphs with unary relation identifying bipartition andwith convex linear order are Ramsey.

Ramsey classes are amalgamation classes

Definition (Amalgamation property of class K)

Nešetril, 80’s: Under mild assumptions Ramsey classes haveamalgamation property.

Ramsey classes are amalgamation classes

Definition (Amalgamation property of class K)

Nešetril, 80’s: Under mild assumptions Ramsey classes haveamalgamation property.

Classification programme

Ramsey classes =⇒ amalgamation classes⇑ ⇓

lifts of homogeneous ⇐= homogeneous structures

Many amalgamation classes are given by the classificationprogramme of homogeneous structures.Can we always find a Ramsey lift?

Theorem (Nešetril, 1989)All homogeneous graphs have Ramsey lift.

Theorem (Jasinski,Laflamme,Nguyen Van Thé,Woodrow, 2014)

All homogeneous digraphs have Ramsey lift.

Our focus: Ramsey lifts of some ages of ω-categoricalstructures

Structures with forbidden homomorphisms

Let F be a family of relational structures. We denote byForbH(F) the class of all finite structures A such that there is noF ∈ F ,F→ A.

Theorem (Cherlin,Shelah,Shi 1998)For every finite family F of finite connected relational structuresthere is an ω-categorical structure that is universal for ForbH(F).

Every ω-categorical structure can be lifted tohomogeneous.Explicit homogenization is given by H. and Nešetril (2009).

Theorem (Nešetril, 2010)

For every finite F there is a Ramsey lift of ForbH(F).

Infinite families of forbidden substructures

Definition(Possibly infinite) family F is regularizable if there isω-categorical structure that is universal for ForbH(F).

. . . and we have an explicit description of such classesgeneralizing the notion of regular languages.

DefinitionClass F (of relational structures) is locally finite in class C if forevery A ∈ C there is only finitely many structures F ∈ F , F→ A.

Theorem (H., Nešetril, 2014)Let E be a family of complete ordered structures, F be a regularfamily of connected structures. Assume that F is locally finite inForbE(E). Then class ForbE(E) ∩ ForbH(F) has Ramsey lift.

Example

Theorem (Nešetril Rödl)Partial orders have Ramsey lift.

P = (V ,≤,≺,⊥)

Forbidden complete substructures E :

Forbidden homomorphic images F :

. . .Structures in ForbE(E) ∪ ForbH(F) can be completed into partialorders without affecting existing ≺ and ⊥ relations.

Example

P = (V ,≤,≺,⊥)Forbidden complete substructures E :

Example

Structures in ForbE(E) ∪ ForbH(F) can be completed into partialorders without affecting existing ≺ and ⊥ relations.

Example

Other Corollaries

Metric spaces with bounded discrete metric

(by forbidding cycles that violate triangular inequality)

Metric spaces (Nešetril, 2005)

(for every given A and B it translates to previous case)

Graph metric spaces . . . and subclases.

Can we handle the complete catalogue?

Forbidden odd cycles up to given a length L.

All finite families F are regular.

This does not generalize to bipartite graphs!

Other Corollaries

This does not generalize to bipartite graphs!J. Hubicka Ramsey classes with algebraical closure and forbidden homomorphisms

Classes with closures

Theorem (Cherlin,Shelah,Shi 1998)LetM be finite family of finite connected structures. There isω-categorical universal structure for ForbM(M) iff if thealgebraic closure is locally finite.

ForbM(M) is a class of all finite structures A such that there isno monomorphism from E ∈ E to A.

The easiest example of such class is the class of bow-tie-freegraphs. Does it have Ramsey expansion?

Classes with closures

Theorem (Cherlin,Shelah,Shi 1998)LetM be finite family of finite connected structures. There isω-categorical universal structure for ForbM(M) iff if thealgebraic closure is locally finite.

ForbM(M) is a class of all finite structures A such that there isno monomorphism from E ∈ E to A.

The easiest example of such class is the class of bow-tie-freegraphs. Does it have Ramsey expansion?

Unary closures

We extend the language by binary relation C representingclosure edges.The unary closure description U is a set of permitted (open)C-out-neighborhoods with only one possible neighborhood forevery substructure on 1 vertex.

DefinitionWe say that structure A is U-closed if for every vertex v the isstructure induced on its C-out-neighborhood is in U.

Theorem (H., Nešetril, 2014)Let E be a family of complete ordered structure and U an unaryclosure description. Then the class of all U-closed structures inForbE(E) has Ramsey lift.

The class of U-closed structures can be homogenized tohereditary lift (with finitely many new relations) preserving R. p.

Unary closures

Example

Theorem (H., Nešetril, 2014)The class of graphs not containing bow-tie as non-inducedsubgraph have Ramsey lift.

Bow-tie graph:

Amalgamation of two triangles must unify vertices.

Wrong!

Example

Theorem (H., Nešetril, 2014)The class of graphs not containing bow-tie as non-inducedsubgraph have Ramsey lift.

Bow-tie graph:

Amalgamation of two triangles must unify vertices.

Wrong!

Example

Structure of bow-tie-free graphs

Edges in no triangles Edges in 1 triangle Edges in > 1 triangles

Closure of a vertex v = all endpoints of red edges contained intriangles containing v .

LemmaBow-tie-free graphs have free amalgamation over closed structures.

Example

Edges in no triangles Edges in 1 triangle Edges in > 1 trianglesClosure of a vertex v = all endpoints of red edges contained intriangles containing v .

Example

Edges in no triangles Edges in 1 triangle Edges in > 1 trianglesClosure of a vertex v = all endpoints of red edges contained intriangles containing v .

Example

Closure description U:

By E describe 3-edge-colored and 7-vertex-colored graph withclosure edges and forbid all triangles except for B-B-R andR-R-R.

Example

Closure description U:

By E describe 3-edge-colored and 7-vertex-colored graph withclosure edges and forbid all triangles except for B-B-R andR-R-R.

Further applications

Cherlin-Shelah-Shi classes with single constraint(ForbM(F) where |F| = 1)

Can each of these classes be described as a freeamalgamation class over unary closure?

Semigeneric homogeneous digraph

An interesting case where fictive closure needs to beadded to produce Ramsey lift.

Line graphs

Some non-unary closures can be expanded to unaryclosures.

Line graphs

Thank you. . .

ramsey classes with algebraical closure and forbidden ......examples of ramsey classes example the...

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