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The 42 reducts of the random ordered graph Michael Pinsker Technische Universitt Wien / Universit Diderot - Paris 7 4th Novi Sad Algebraic Conference, 2013 42 Michael Pinsker Outline Part I: The setting of The Answer Part II: The 42

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The 42 reducts of the random ordered graph

Michael Pinsker

Technische Universität Wien / Université Diderot - Paris 7

4th Novi Sad Algebraic Conference, 2013

42 Michael Pinsker

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Outline

Part I: The setting of The Answer Part II: The 42 reducts of the random ordered graph Part III: The effect of The Answer Part IV: The question to The Answer

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Part I: The setting of The Answer

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Homogeneous structures

Let ∆ be a countable structure.

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Homogeneous structures

Let ∆ be a countable structure. Definition ∆ is homogeneous :↔ every isomorphism between finitely generated substructures of ∆ extends to an automorphism of ∆.

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Homogeneous structures

Let ∆ be a countable structure. Definition ∆ is homogeneous :↔ every isomorphism between finitely generated substructures of ∆ extends to an automorphism of ∆. Examples

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Homogeneous structures

Let ∆ be a countable structure. Definition ∆ is homogeneous :↔ every isomorphism between finitely generated substructures of ∆ extends to an automorphism of ∆. Examples Order of the rationals (Q; <)

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Homogeneous structures

Let ∆ be a countable structure. Definition ∆ is homogeneous :↔ every isomorphism between finitely generated substructures of ∆ extends to an automorphism of ∆. Examples Order of the rationals (Q; <) Random graph (V; E)

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Homogeneous structures

Let ∆ be a countable structure. Definition ∆ is homogeneous :↔ every isomorphism between finitely generated substructures of ∆ extends to an automorphism of ∆. Examples Order of the rationals (Q; <) Random graph (V; E) Free Boolean algebra with ℵ0 generators

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Fraïssé limits

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Fraïssé limits

Let C be a class of finitely generated structures in a countable language, closed under isomorphism.

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Fraïssé limits

Let C be a class of finitely generated structures in a countable language, closed under isomorphism. Theorem (Fraïssé) Assume C

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Fraïssé limits

Let C be a class of finitely generated structures in a countable language, closed under isomorphism. Theorem (Fraïssé) Assume C is closed under substructures

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Fraïssé limits

Let C be a class of finitely generated structures in a countable language, closed under isomorphism. Theorem (Fraïssé) Assume C is closed under substructures has joint embeddings: for all B, C ∈ C there is D ∈ C containing isomorphic copies of B, C

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Fraïssé limits

Let C be a class of finitely generated structures in a countable language, closed under isomorphism. Theorem (Fraïssé) Assume C is closed under substructures has joint embeddings: for all B, C ∈ C there is D ∈ C containing isomorphic copies of B, C has amalgamation: for all A, B, C ∈ C and embeddings eB : A → B and eC : A → C there is D ∈ C and embeddings fB : B → D and fC : C → D such that fB ◦ eB = fC ◦ eC.

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Fraïssé limits

Let C be a class of finitely generated structures in a countable language, closed under isomorphism. Theorem (Fraïssé) Assume C is closed under substructures has joint embeddings: for all B, C ∈ C there is D ∈ C containing isomorphic copies of B, C has amalgamation: for all A, B, C ∈ C and embeddings eB : A → B and eC : A → C there is D ∈ C and embeddings fB : B → D and fC : C → D such that fB ◦ eB = fC ◦ eC. Then there exists a unique countable homogeneous structure ∆ whose age (=substructures up to iso) equals C.

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Amalgamation

A

D C B

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Examples

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Examples

(Finite) linear orders ↔ (Q; <)

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Examples

(Finite) linear orders ↔ (Q; <) Undirected graphs ↔ random graph (V; E)

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Examples

(Finite) linear orders ↔ (Q; <) Undirected graphs ↔ random graph (V; E) Boolean algebras ↔ random (= free) Boolean algebra

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Examples

(Finite) linear orders ↔ (Q; <) Undirected graphs ↔ random graph (V; E) Boolean algebras ↔ random (= free) Boolean algebra Lattices ↔ random lattice

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Examples

(Finite) linear orders ↔ (Q; <) Undirected graphs ↔ random graph (V; E) Boolean algebras ↔ random (= free) Boolean algebra Lattices ↔ random lattice Distributive lattices ↔ random distributive lattice

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Examples

(Finite) linear orders ↔ (Q; <) Undirected graphs ↔ random graph (V; E) Boolean algebras ↔ random (= free) Boolean algebra Lattices ↔ random lattice Distributive lattices ↔ random distributive lattice Partial orders ↔ random partial order

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Examples

(Finite) linear orders ↔ (Q; <) Undirected graphs ↔ random graph (V; E) Boolean algebras ↔ random (= free) Boolean algebra Lattices ↔ random lattice Distributive lattices ↔ random distributive lattice Partial orders ↔ random partial order Tournaments ↔ random tournament

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Examples

(Finite) linear orders ↔ (Q; <) Undirected graphs ↔ random graph (V; E) Boolean algebras ↔ random (= free) Boolean algebra Lattices ↔ random lattice Distributive lattices ↔ random distributive lattice Partial orders ↔ random partial order Tournaments ↔ random tournament Linearly ordered graphs ↔ random ordered graph (D; <, E)

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Reducts

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Reducts

Let ∆ be a structure.

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Reducts

Let ∆ be a structure. Definition A reduct of ∆ is a structure on the same domain whose relations and functions are first-order definable in ∆ (without parameters).

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Reducts

Let ∆ be a structure. Definition A reduct of ∆ is a structure on the same domain whose relations and functions are first-order definable in ∆ (without parameters). Examples

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Reducts

Let ∆ be a structure. Definition A reduct of ∆ is a structure on the same domain whose relations and functions are first-order definable in ∆ (without parameters). Examples (Q; <): reduct (Q; Between(x, y, z))

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Reducts

Let ∆ be a structure. Definition A reduct of ∆ is a structure on the same domain whose relations and functions are first-order definable in ∆ (without parameters). Examples (Q; <): reduct (Q; Between(x, y, z)) (Q; <): reduct (Q; >)

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Reducts

Let ∆ be a structure. Definition A reduct of ∆ is a structure on the same domain whose relations and functions are first-order definable in ∆ (without parameters). Examples (Q; <): reduct (Q; Between(x, y, z)) (Q; <): reduct (Q; >) random graph (V; E): reduct (V; K3(x, y, z))

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Reducts

Let ∆ be a structure. Definition A reduct of ∆ is a structure on the same domain whose relations and functions are first-order definable in ∆ (without parameters). Examples (Q; <): reduct (Q; Between(x, y, z)) (Q; <): reduct (Q; >) random graph (V; E): reduct (V; K3(x, y, z)) random poset (P; ≤): reduct (P; ⊥(x, y))

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Reducts

Let ∆ be a structure. Definition A reduct of ∆ is a structure on the same domain whose relations and functions are first-order definable in ∆ (without parameters). Examples (Q; <): reduct (Q; Between(x, y, z)) (Q; <): reduct (Q; >) random graph (V; E): reduct (V; K3(x, y, z)) random poset (P; ≤): reduct (P; ⊥(x, y)) Problem Understand the reducts of homogeneous structures.

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Motivation

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Motivation

Why reducts?

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Motivation

Why reducts? Understand ∆ itself:

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Motivation

Why reducts? Understand ∆ itself: – its first-order theory

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Motivation

Why reducts? Understand ∆ itself: – its first-order theory – its symmetries (via connection with permutation groups)

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Motivation

Why reducts? Understand ∆ itself: – its first-order theory – its symmetries (via connection with permutation groups) Understand the age C of ∆:

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Motivation

Why reducts? Understand ∆ itself: – its first-order theory – its symmetries (via connection with permutation groups) Understand the age C of ∆: – uniform group actions on C (via permutation groups - combinatorics of C)

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Motivation

Why reducts? Understand ∆ itself: – its first-order theory – its symmetries (via connection with permutation groups) Understand the age C of ∆: – uniform group actions on C (via permutation groups - combinatorics of C) – Constraint Satisfaction Problems related to C: Graph-SAT, Poset-SAT,. . .

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Reducts up to first-order equivalence

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Reducts up to first-order equivalence

For reducts Γ, Γ′ of ∆ set Γ ≤ Γ′ iff Γ is a reduct of Γ′.

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Reducts up to first-order equivalence

For reducts Γ, Γ′ of ∆ set Γ ≤ Γ′ iff Γ is a reduct of Γ′. Quasiorder.

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Reducts up to first-order equivalence

For reducts Γ, Γ′ of ∆ set Γ ≤ Γ′ iff Γ is a reduct of Γ′. Quasiorder. Consider reducts Γ, Γ′ equivalent iff Γ ≤ Γ′ and Γ′ ≤ Γ.

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Reducts up to first-order equivalence

For reducts Γ, Γ′ of ∆ set Γ ≤ Γ′ iff Γ is a reduct of Γ′. Quasiorder. Consider reducts Γ, Γ′ equivalent iff Γ ≤ Γ′ and Γ′ ≤ Γ. Factoring out we get a complete lattice.

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Reducts up to first-order equivalence

For reducts Γ, Γ′ of ∆ set Γ ≤ Γ′ iff Γ is a reduct of Γ′. Quasiorder. Consider reducts Γ, Γ′ equivalent iff Γ ≤ Γ′ and Γ′ ≤ Γ. Factoring out we get a complete lattice. Multiple choice: Equivalent or not?

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Reducts up to first-order equivalence

For reducts Γ, Γ′ of ∆ set Γ ≤ Γ′ iff Γ is a reduct of Γ′. Quasiorder. Consider reducts Γ, Γ′ equivalent iff Γ ≤ Γ′ and Γ′ ≤ Γ. Factoring out we get a complete lattice. Multiple choice: Equivalent or not? (Q; <) and (Q; >)

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Reducts up to first-order equivalence

For reducts Γ, Γ′ of ∆ set Γ ≤ Γ′ iff Γ is a reduct of Γ′. Quasiorder. Consider reducts Γ, Γ′ equivalent iff Γ ≤ Γ′ and Γ′ ≤ Γ. Factoring out we get a complete lattice. Multiple choice: Equivalent or not? (Q; <) and (Q; >) (Q; <) and (Q; Between(x, y, z))

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Reducts up to first-order equivalence

For reducts Γ, Γ′ of ∆ set Γ ≤ Γ′ iff Γ is a reduct of Γ′. Quasiorder. Consider reducts Γ, Γ′ equivalent iff Γ ≤ Γ′ and Γ′ ≤ Γ. Factoring out we get a complete lattice. Multiple choice: Equivalent or not? (Q; <) and (Q; >) (Q; <) and (Q; Between(x, y, z)) random poset (P; ≤) and (P; ⊥(x, y))

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Reducts up to first-order equivalence

For reducts Γ, Γ′ of ∆ set Γ ≤ Γ′ iff Γ is a reduct of Γ′. Quasiorder. Consider reducts Γ, Γ′ equivalent iff Γ ≤ Γ′ and Γ′ ≤ Γ. Factoring out we get a complete lattice. Multiple choice: Equivalent or not? (Q; <) and (Q; >) (Q; <) and (Q; Between(x, y, z)) random poset (P; ≤) and (P; ⊥(x, y)) random graph (V; E) and (V; K3(x, y, z))

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Reducts up to first-order equivalence

For reducts Γ, Γ′ of ∆ set Γ ≤ Γ′ iff Γ is a reduct of Γ′. Quasiorder. Consider reducts Γ, Γ′ equivalent iff Γ ≤ Γ′ and Γ′ ≤ Γ. Factoring out we get a complete lattice. Multiple choice: Equivalent or not? (Q; <) and (Q; >) (Q; <) and (Q; Between(x, y, z)) random poset (P; ≤) and (P; ⊥(x, y)) random graph (V; E) and (V; K3(x, y, z)) Question How many inequivalent reducts?

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Examples

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Examples

(Q; <): 5 (Cameron ’76)

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Examples

(Q; <): 5 (Cameron ’76) random graph (V; E): 5 (Thomas ’91)

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Examples

(Q; <): 5 (Cameron ’76) random graph (V; E): 5 (Thomas ’91) random k-hypergraph: 2k + 1 (Thomas ’96)

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Examples

(Q; <): 5 (Cameron ’76) random graph (V; E): 5 (Thomas ’91) random k-hypergraph: 2k + 1 (Thomas ’96) random tournament: 5 (Bennett ’97)

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Examples

(Q; <): 5 (Cameron ’76) random graph (V; E): 5 (Thomas ’91) random k-hypergraph: 2k + 1 (Thomas ’96) random tournament: 5 (Bennett ’97) (Q; <, 0): 116 (Junker+Ziegler ’08)

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Examples

(Q; <): 5 (Cameron ’76) random graph (V; E): 5 (Thomas ’91) random k-hypergraph: 2k + 1 (Thomas ’96) random tournament: 5 (Bennett ’97) (Q; <, 0): 116 (Junker+Ziegler ’08) random partial order: 5 (Pach+MP+Pongrácz+Szabó ’11)

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Examples

(Q; <): 5 (Cameron ’76) random graph (V; E): 5 (Thomas ’91) random k-hypergraph: 2k + 1 (Thomas ’96) random tournament: 5 (Bennett ’97) (Q; <, 0): 116 (Junker+Ziegler ’08) random partial order: 5 (Pach+MP+Pongrácz+Szabó ’11) Conjecture (Thomas ’91) Homogeneous structures in finite relational language have finitely many reducts.

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Permutation groups

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Permutation groups

A permutation group is closed :↔ it contains all permutations which it can interpolate on finite subsets.

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Permutation groups

A permutation group is closed :↔ it contains all permutations which it can interpolate on finite subsets. Theorem (Corollary of Ryll-Nardzewski, Engeler, Svenonius) Let ∆ be homogeneous in a finite relational language. Then the mapping Γ → Aut(Γ)

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Permutation groups

A permutation group is closed :↔ it contains all permutations which it can interpolate on finite subsets. Theorem (Corollary of Ryll-Nardzewski, Engeler, Svenonius) Let ∆ be homogeneous in a finite relational language. Then the mapping Γ → Aut(Γ) is an anti-isomorphism from the lattice of reducts to the lattice of closed supergroups of Aut(∆).

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The rationals (Q; <)

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The rationals (Q; <)

Let ↔ be any permutation of Q which reverses the order.

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The rationals (Q; <)

Let ↔ be any permutation of Q which reverses the order. Let be any permutation of Q which for some irrational π puts (−∞; π) behind (π; ∞) and preserves the order otherwise.

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The rationals (Q; <)

Let ↔ be any permutation of Q which reverses the order. Let be any permutation of Q which for some irrational π puts (−∞; π) behind (π; ∞) and preserves the order otherwise. Theorem (Cameron ’76) The closed supergroups of Aut(Q; <) are precisely:

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The rationals (Q; <)

Let ↔ be any permutation of Q which reverses the order. Let be any permutation of Q which for some irrational π puts (−∞; π) behind (π; ∞) and preserves the order otherwise. Theorem (Cameron ’76) The closed supergroups of Aut(Q; <) are precisely: Aut(Q; <)

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The rationals (Q; <)

Let ↔ be any permutation of Q which reverses the order. Let be any permutation of Q which for some irrational π puts (−∞; π) behind (π; ∞) and preserves the order otherwise. Theorem (Cameron ’76) The closed supergroups of Aut(Q; <) are precisely: Aut(Q; <) {↔} ∪ Aut(Q; <)

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The rationals (Q; <)

Let ↔ be any permutation of Q which reverses the order. Let be any permutation of Q which for some irrational π puts (−∞; π) behind (π; ∞) and preserves the order otherwise. Theorem (Cameron ’76) The closed supergroups of Aut(Q; <) are precisely: Aut(Q; <) {↔} ∪ Aut(Q; <) {} ∪ Aut(Q; <)

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The rationals (Q; <)

Let ↔ be any permutation of Q which reverses the order. Let be any permutation of Q which for some irrational π puts (−∞; π) behind (π; ∞) and preserves the order otherwise. Theorem (Cameron ’76) The closed supergroups of Aut(Q; <) are precisely: Aut(Q; <) {↔} ∪ Aut(Q; <) {} ∪ Aut(Q; <) {↔, } ∪ Aut(Q; <)

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The rationals (Q; <)

Let ↔ be any permutation of Q which reverses the order. Let be any permutation of Q which for some irrational π puts (−∞; π) behind (π; ∞) and preserves the order otherwise. Theorem (Cameron ’76) The closed supergroups of Aut(Q; <) are precisely: Aut(Q; <) {↔} ∪ Aut(Q; <) {} ∪ Aut(Q; <) {↔, } ∪ Aut(Q; <) Sym(Q)

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The rationals (Q; <)

Let ↔ be any permutation of Q which reverses the order. Let be any permutation of Q which for some irrational π puts (−∞; π) behind (π; ∞) and preserves the order otherwise. Theorem (Cameron ’76) The closed supergroups of Aut(Q; <) are precisely: Aut(Q; <) {↔} ∪ Aut(Q; <) = Aut(Q; Between(x, y, z)) {} ∪ Aut(Q; <) {↔, } ∪ Aut(Q; <) Sym(Q)

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The rationals (Q; <)

Let ↔ be any permutation of Q which reverses the order. Let be any permutation of Q which for some irrational π puts (−∞; π) behind (π; ∞) and preserves the order otherwise. Theorem (Cameron ’76) The closed supergroups of Aut(Q; <) are precisely: Aut(Q; <) {↔} ∪ Aut(Q; <) = Aut(Q; Between(x, y, z)) {} ∪ Aut(Q; <) = Aut(Q; Cyclic(x, y, z)) {↔, } ∪ Aut(Q; <) Sym(Q)

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The rationals (Q; <)

Let ↔ be any permutation of Q which reverses the order. Let be any permutation of Q which for some irrational π puts (−∞; π) behind (π; ∞) and preserves the order otherwise. Theorem (Cameron ’76) The closed supergroups of Aut(Q; <) are precisely: Aut(Q; <) {↔} ∪ Aut(Q; <) = Aut(Q; Between(x, y, z)) {} ∪ Aut(Q; <) = Aut(Q; Cyclic(x, y, z)) {↔, } ∪ Aut(Q; <) = Aut(Q; Separate(x, y, u, v)) Sym(Q)

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The random graph (V; E)

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges.

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges. Let sw be any permutation which for some finite A ⊆ V switches edges and non-edges between A and V \ A and preserves the graph relation on A and V \ A.

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges. Let sw be any permutation which for some finite A ⊆ V switches edges and non-edges between A and V \ A and preserves the graph relation on A and V \ A. Theorem (Thomas ’91) The closed supergroups of Aut(V; E) are precisely:

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges. Let sw be any permutation which for some finite A ⊆ V switches edges and non-edges between A and V \ A and preserves the graph relation on A and V \ A. Theorem (Thomas ’91) The closed supergroups of Aut(V; E) are precisely: Aut(V; E)

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges. Let sw be any permutation which for some finite A ⊆ V switches edges and non-edges between A and V \ A and preserves the graph relation on A and V \ A. Theorem (Thomas ’91) The closed supergroups of Aut(V; E) are precisely: Aut(V; E) {sw} ∪ Aut(V; E)

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges. Let sw be any permutation which for some finite A ⊆ V switches edges and non-edges between A and V \ A and preserves the graph relation on A and V \ A. Theorem (Thomas ’91) The closed supergroups of Aut(V; E) are precisely: Aut(V; E) {sw} ∪ Aut(V; E) {−} ∪ Aut(V; E)

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges. Let sw be any permutation which for some finite A ⊆ V switches edges and non-edges between A and V \ A and preserves the graph relation on A and V \ A. Theorem (Thomas ’91) The closed supergroups of Aut(V; E) are precisely: Aut(V; E) {sw} ∪ Aut(V; E) {−} ∪ Aut(V; E) {−, sw} ∪ Aut(V; E)

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges. Let sw be any permutation which for some finite A ⊆ V switches edges and non-edges between A and V \ A and preserves the graph relation on A and V \ A. Theorem (Thomas ’91) The closed supergroups of Aut(V; E) are precisely: Aut(V; E) {sw} ∪ Aut(V; E) {−} ∪ Aut(V; E) {−, sw} ∪ Aut(V; E) Sym(V)

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges. Let sw be any permutation which for some finite A ⊆ V switches edges and non-edges between A and V \ A and preserves the graph relation on A and V \ A. Theorem (Thomas ’91) The closed supergroups of Aut(V; E) are precisely: Aut(V; E) {sw} ∪ Aut(V; E) {−} ∪ Aut(V; E) {−, sw} ∪ Aut(V; E) Sym(V) For k ≥ 1, let R(k) consist of the k-tuples of distinct elements of V which induce an odd number of edges.

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges. Let sw be any permutation which for some finite A ⊆ V switches edges and non-edges between A and V \ A and preserves the graph relation on A and V \ A. Theorem (Thomas ’91) The closed supergroups of Aut(V; E) are precisely: Aut(V; E) {sw} ∪ Aut(V; E) = Aut(V; R(3)) {−} ∪ Aut(V; E) {−, sw} ∪ Aut(V; E) Sym(V) For k ≥ 1, let R(k) consist of the k-tuples of distinct elements of V which induce an odd number of edges.

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges. Let sw be any permutation which for some finite A ⊆ V switches edges and non-edges between A and V \ A and preserves the graph relation on A and V \ A. Theorem (Thomas ’91) The closed supergroups of Aut(V; E) are precisely: Aut(V; E) {sw} ∪ Aut(V; E) = Aut(V; R(3)) {−} ∪ Aut(V; E) = Aut(V; R(4)) {−, sw} ∪ Aut(V; E) Sym(V) For k ≥ 1, let R(k) consist of the k-tuples of distinct elements of V which induce an odd number of edges.

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The random graph (V; E)

Let − be any permutation of V which switches edges and non-edges. Let sw be any permutation which for some finite A ⊆ V switches edges and non-edges between A and V \ A and preserves the graph relation on A and V \ A. Theorem (Thomas ’91) The closed supergroups of Aut(V; E) are precisely: Aut(V; E) {sw} ∪ Aut(V; E) = Aut(V; R(3)) {−} ∪ Aut(V; E) = Aut(V; R(4)) {−, sw} ∪ Aut(V; E) = Aut(V; R(5)) Sym(V) For k ≥ 1, let R(k) consist of the k-tuples of distinct elements of V which induce an odd number of edges.

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Part II: The 42 reducts of the random ordered graph

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The random ordered graph

Definition The random ordered graph (D; <, E) is the unique countable linearly ordered graph which contains all finite linearly ordered graphs is homogeneous.

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The random ordered graph

Definition The random ordered graph (D; <, E) is the unique countable linearly ordered graph which contains all finite linearly ordered graphs is homogeneous. Observation (D; <) is the order of the rationals (D; E) is the random graph

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The random ordered graph

Definition The random ordered graph (D; <, E) is the unique countable linearly ordered graph which contains all finite linearly ordered graphs is homogeneous. Observation (D; <) is the order of the rationals (D; E) is the random graph This is because the two structures are superposed freely, i.e., in all possible ways.

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Strong amalgamation

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Strong amalgamation

Definition A class C has strong amalgamation :↔ for all A, B, C ∈ C and embeddings eB : A → B and eC : A → C there is D ∈ C and embeddings fB : B → D and fC : C → D such that fB ◦ eB = fC ◦ eC and fB[B] ∩ fC[C] = fB ◦ eB[A].

A D C B

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Mixing

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Mixing

Let τ1, τ2 be disjoint languages. Let C1, C2 Fraïssé classes in those languages, ∆1, ∆2 be their limits.

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Mixing

Let τ1, τ2 be disjoint languages. Let C1, C2 Fraïssé classes in those languages, ∆1, ∆2 be their limits. Free superposition Assume that C1, C2 have strong amalgamation.

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Mixing

Let τ1, τ2 be disjoint languages. Let C1, C2 Fraïssé classes in those languages, ∆1, ∆2 be their limits. Free superposition Assume that C1, C2 have strong amalgamation. Then the class C of τ1 ∪ τ2-structures whose τi-reduct is in Ci

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Mixing

Let τ1, τ2 be disjoint languages. Let C1, C2 Fraïssé classes in those languages, ∆1, ∆2 be their limits. Free superposition Assume that C1, C2 have strong amalgamation. Then the class C of τ1 ∪ τ2-structures whose τi-reduct is in Ci is a Fraïssé class and the τi-reduct of its limit is isomorphic to ∆i.

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Trivial reducts of the random ordered graph

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Trivial reducts of the random ordered graph

Every reduct of (D; <) is a reduct of the random ordered graph.

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Trivial reducts of the random ordered graph

Every reduct of (D; <) is a reduct of the random ordered graph. Every reduct of (D; E) is a reduct of the random ordered graph.

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Trivial reducts of the random ordered graph

Every reduct of (D; <) is a reduct of the random ordered graph. Every reduct of (D; E) is a reduct of the random ordered graph. If (D; R) is a reduct of (D; <) and (D; S) is a reduct of (D; E) then (D; R, S) is a reduct of the random ordered graph.

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Trivial reducts of the random ordered graph

Every reduct of (D; <) is a reduct of the random ordered graph. Every reduct of (D; E) is a reduct of the random ordered graph. If (D; R) is a reduct of (D; <) and (D; S) is a reduct of (D; E) then (D; R, S) is a reduct of the random ordered graph. Corresponds to intersecting the groups Aut(D; R) and Aut(D; S).

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Trivial reducts of the random ordered graph

Every reduct of (D; <) is a reduct of the random ordered graph. Every reduct of (D; E) is a reduct of the random ordered graph. If (D; R) is a reduct of (D; <) and (D; S) is a reduct of (D; E) then (D; R, S) is a reduct of the random ordered graph. Corresponds to intersecting the groups Aut(D; R) and Aut(D; S). Yields distinct reducts because of free superposition.

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Trivial reducts of the random ordered graph

Every reduct of (D; <) is a reduct of the random ordered graph. Every reduct of (D; E) is a reduct of the random ordered graph. If (D; R) is a reduct of (D; <) and (D; S) is a reduct of (D; E) then (D; R, S) is a reduct of the random ordered graph. Corresponds to intersecting the groups Aut(D; R) and Aut(D; S). Yields distinct reducts because of free superposition. Examples

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Trivial reducts of the random ordered graph

Every reduct of (D; <) is a reduct of the random ordered graph. Every reduct of (D; E) is a reduct of the random ordered graph. If (D; R) is a reduct of (D; <) and (D; S) is a reduct of (D; E) then (D; R, S) is a reduct of the random ordered graph. Corresponds to intersecting the groups Aut(D; R) and Aut(D; S). Yields distinct reducts because of free superposition. Examples Keeping the order while flipping the graph relation.

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Trivial reducts of the random ordered graph

Every reduct of (D; <) is a reduct of the random ordered graph. Every reduct of (D; E) is a reduct of the random ordered graph. If (D; R) is a reduct of (D; <) and (D; S) is a reduct of (D; E) then (D; R, S) is a reduct of the random ordered graph. Corresponds to intersecting the groups Aut(D; R) and Aut(D; S). Yields distinct reducts because of free superposition. Examples Keeping the order while flipping the graph relation. Reversing the order while keeping the graph relation.

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Trivial reducts of the random ordered graph

Every reduct of (D; <) is a reduct of the random ordered graph. Every reduct of (D; E) is a reduct of the random ordered graph. If (D; R) is a reduct of (D; <) and (D; S) is a reduct of (D; E) then (D; R, S) is a reduct of the random ordered graph. Corresponds to intersecting the groups Aut(D; R) and Aut(D; S). Yields distinct reducts because of free superposition. Examples Keeping the order while flipping the graph relation. Reversing the order while keeping the graph relation. Lemma The random ordered graph has at least 25 reducts.

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Similarities between the order and the graph reducts

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Similarities between the order and the graph reducts

The following permutations yield new non-trivial reducts.

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Similarities between the order and the graph reducts

The following permutations yield new non-trivial reducts. reversing the order and simultaneously flipping the graph relation

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Similarities between the order and the graph reducts

The following permutations yield new non-trivial reducts. reversing the order and simultaneously flipping the graph relation for an irrational π, put (−∞, π) behind (π, ∞) whilst flipping the graph relation between these parts.

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Similarities between the order and the graph reducts

The following permutations yield new non-trivial reducts. reversing the order and simultaneously flipping the graph relation for an irrational π, put (−∞, π) behind (π, ∞) whilst flipping the graph relation between these parts. No other combination of this kind!

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Similarities between the order and the graph reducts

The following permutations yield new non-trivial reducts. reversing the order and simultaneously flipping the graph relation for an irrational π, put (−∞, π) behind (π, ∞) whilst flipping the graph relation between these parts. No other combination of this kind! Lemma The random ordered graph has at least 27 reducts.

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Hello random tournament!

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Hello random tournament!

Definition A tournament is a digraph with precisely one edge between any two vertices.

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Hello random tournament!

Definition A tournament is a digraph with precisely one edge between any two vertices. Theorem (Bennett ’97) The random tournament has 5 reducts.

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Hello random tournament!

Definition A tournament is a digraph with precisely one edge between any two vertices. Theorem (Bennett ’97) The random tournament has 5 reducts. Observation Set T(x, y) iff x < y ∧ E(x, y) or x > y ∧ N(x, y). Then (D; T) is the random tournament.

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Hello random tournament!

Definition A tournament is a digraph with precisely one edge between any two vertices. Theorem (Bennett ’97) The random tournament has 5 reducts. Observation Set T(x, y) iff x < y ∧ E(x, y) or x > y ∧ N(x, y). Then (D; T) is the random tournament. Lemma The random ordered graph has at least 32 reducts.

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Finally, some asymmetry

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Finally, some asymmetry

The following permutations yield new non-trivial reducts.

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Finally, some asymmetry

The following permutations yield new non-trivial reducts. preserving the order whilst flipping the graph relation below some irrational.

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Finally, some asymmetry

The following permutations yield new non-trivial reducts. preserving the order whilst flipping the graph relation below some irrational. preserving the order whilst flipping the graph relation above some irrational.

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Finally, some asymmetry

The following permutations yield new non-trivial reducts. preserving the order whilst flipping the graph relation below some irrational. preserving the order whilst flipping the graph relation above some irrational. There are no “dual” permutations of these.

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Finally, some asymmetry

The following permutations yield new non-trivial reducts. preserving the order whilst flipping the graph relation below some irrational. preserving the order whilst flipping the graph relation above some irrational. There are no “dual” permutations of these. Lemma The random ordered graph has at least 32 reducts.

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Theorem (Bodirsky+MP+Pongrácz ’13) The random ordered graph has 41 reducts.

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Theorem (Bodirsky+MP+Pongrácz ’13) The random ordered graph has 41 reducts.

1 4 5 3 2 6 7 9 11 13 10 12 14 8 18 15 16 17 27 23 24 25 26 28 29 30 31 32 19 20 21 22 34 33 35 36 37 38 39 40 41 Aut(E,<) (sw,turn) (-,lr) (id,lr) (id,turn) (sw,id) (-,id) Aut(E,sep) Aut(E) Aut(<) Aut(sep) Aut(R(5)) Aut(R(5),<) Aut(T)

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Part III: The effect of The Answer

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Discussion

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Discussion

We have learnt from the result:

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Discussion

We have learnt from the result: similarities between the symmetries of the order and the graph

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Discussion

We have learnt from the result: similarities between the symmetries of the order and the graph nonetheless their combination yields an asymmetry

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Discussion

We have learnt from the result: similarities between the symmetries of the order and the graph nonetheless their combination yields an asymmetry we cannot calculate the reducts of a superposed structure from its factors

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Discussion

We have learnt from the result: similarities between the symmetries of the order and the graph nonetheless their combination yields an asymmetry we cannot calculate the reducts of a superposed structure from its factors On a technical level:

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Discussion

We have learnt from the result: similarities between the symmetries of the order and the graph nonetheless their combination yields an asymmetry we cannot calculate the reducts of a superposed structure from its factors On a technical level:

  • ur Ramsey-theoretic method is quite efficient

(first classification of free superposition)

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Discussion

We have learnt from the result: similarities between the symmetries of the order and the graph nonetheless their combination yields an asymmetry we cannot calculate the reducts of a superposed structure from its factors On a technical level:

  • ur Ramsey-theoretic method is quite efficient

(first classification of free superposition) improved it to reduce work to the join irreducible elements

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Discussion

We have learnt from the result: similarities between the symmetries of the order and the graph nonetheless their combination yields an asymmetry we cannot calculate the reducts of a superposed structure from its factors On a technical level:

  • ur Ramsey-theoretic method is quite efficient

(first classification of free superposition) improved it to reduce work to the join irreducible elements

  • ur method is not sporadic (same for order, graph, tournament)

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Ramsey structures

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Ramsey structures

Definition (Ramsey structure ∆)

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Ramsey structures

Definition (Ramsey structure ∆) For all finite substructures P, H of ∆: Whenever we color the copies of P in ∆ with 2 colors then there is a monochromatic copy of H in ∆.

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Ramsey structures

Definition (Ramsey structure ∆) For all finite substructures P, H of ∆: Whenever we color the copies of P in ∆ with 2 colors then there is a monochromatic copy of H in ∆.

P P

Δ

H

P P P P P P P P

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Ramsey structures

Definition (Ramsey structure ∆) For all finite substructures P, H of ∆: Whenever we color the copies of P in ∆ with 2 colors then there is a monochromatic copy of H in ∆.

P P

Δ

H

P P P P P P P P

Theorem (Nešetˇ ril-Rödl) The random ordered graph is Ramsey.

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Canonical functions

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Canonical functions

Definition Let ∆, Λ be structures. f : ∆ → Λ is canonical iff for all tuples (x1, . . . , xn), (y1, . . . , yn) of the same type in ∆ (f(x1), . . . , f(xn)) and (f(y1), . . . , f(yn)) have the same type in Λ.

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Canonical functions

Definition Let ∆, Λ be structures. f : ∆ → Λ is canonical iff for all tuples (x1, . . . , xn), (y1, . . . , yn) of the same type in ∆ (f(x1), . . . , f(xn)) and (f(y1), . . . , f(yn)) have the same type in Λ. Examples on (D; <, E)

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Canonical functions

Definition Let ∆, Λ be structures. f : ∆ → Λ is canonical iff for all tuples (x1, . . . , xn), (y1, . . . , yn) of the same type in ∆ (f(x1), . . . , f(xn)) and (f(y1), . . . , f(yn)) have the same type in Λ. Examples on (D; <, E) self-embeddings

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Canonical functions

Definition Let ∆, Λ be structures. f : ∆ → Λ is canonical iff for all tuples (x1, . . . , xn), (y1, . . . , yn) of the same type in ∆ (f(x1), . . . , f(xn)) and (f(y1), . . . , f(yn)) have the same type in Λ. Examples on (D; <, E) self-embeddings reversing <, preserving edges and non-edges

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Canonical functions

Definition Let ∆, Λ be structures. f : ∆ → Λ is canonical iff for all tuples (x1, . . . , xn), (y1, . . . , yn) of the same type in ∆ (f(x1), . . . , f(xn)) and (f(y1), . . . , f(yn)) have the same type in Λ. Examples on (D; <, E) self-embeddings reversing <, preserving edges and non-edges preserving <, flipping edges and non-edges

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Canonical functions

Definition Let ∆, Λ be structures. f : ∆ → Λ is canonical iff for all tuples (x1, . . . , xn), (y1, . . . , yn) of the same type in ∆ (f(x1), . . . , f(xn)) and (f(y1), . . . , f(yn)) have the same type in Λ. Examples on (D; <, E) self-embeddings reversing <, preserving edges and non-edges preserving <, flipping edges and non-edges preserving <, send to clique

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Canonizing functions on Ramsey structures

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Canonizing functions on Ramsey structures

Magical proposition (Bodirsky+MP+Tsankov ’11) Let ∆ is ordered Ramsey homogeneous finite language f : ∆ → ∆ c1, . . . , cn ∈ ∆.

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Canonizing functions on Ramsey structures

Magical proposition (Bodirsky+MP+Tsankov ’11) Let ∆ is ordered Ramsey homogeneous finite language f : ∆ → ∆ c1, . . . , cn ∈ ∆. Then the closed monoid generated by {f} ∪ Aut(∆) contains a function g which

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Canonizing functions on Ramsey structures

Magical proposition (Bodirsky+MP+Tsankov ’11) Let ∆ is ordered Ramsey homogeneous finite language f : ∆ → ∆ c1, . . . , cn ∈ ∆. Then the closed monoid generated by {f} ∪ Aut(∆) contains a function g which is canonical as a function from (∆, c1, . . . , cn) to ∆ is identical with f on {c1, . . . , cn}.

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Canonizing functions on Ramsey structures

Magical proposition (Bodirsky+MP+Tsankov ’11) Let ∆ is ordered Ramsey homogeneous finite language f : ∆ → ∆ c1, . . . , cn ∈ ∆. Then the closed monoid generated by {f} ∪ Aut(∆) contains a function g which is canonical as a function from (∆, c1, . . . , cn) to ∆ is identical with f on {c1, . . . , cn}. Note:

  • nly finitely many different behaviors of canonical functions.

g, g′ same behavior → generate one another (with Aut(∆)).

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1 4 5 3 2 6 7 9 11 13 10 12 14 8 18 15 16 17 27 23 24 25 26 28 29 30 31 32 19 20 21 22 34 33 35 36 37 38 39 40 41 Aut(E,<) (sw,turn) (-,lr) (id,lr) (id,turn) (sw,id) (-,id) Aut(E,sep) Aut(E) Aut(<) Aut(sep) Aut(R(5)) Aut(R(5),<) Aut(T)

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Part IV: The Question to The Answer

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The Question

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The Question

Problem Suppose that ∆1, ∆2 have finitely many reducts. Does their free superposition have finitely many reducts?

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The Question

Problem Suppose that ∆1, ∆2 have finitely many reducts. Does their free superposition have finitely many reducts? Problem Suppose that ∆ is homogeneous in a finite relational language. Does it have a finite homogeneous extension which is Ramsey?

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Thank you!

1 4 5 3 2 6 7 9 11 13 10 12 14 8 18 15 16 17 27 23 24 25 26 28 29 30 31 32 19 20 21 22 34 33 35 36 37 38 39 40 41 Aut(E,<) (sw,turn) (-,lr) (id,lr) (id,turn) (sw,id) (-,id) Aut(E,sep) Aut(E) Aut(<) Aut(sep) Aut(R(5)) Aut(R(5),<) Aut(T)

“The Answer to the Great Question. . . Of Life, the Universe and Everything. . . Is. . . Forty-two,” said Deep Thought, with infinite majesty and calm.

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