SãoSebastião,BrasilAugust15,2013 STW2013InhonourofOfeliaAlas PontificiaUniversidadJaverianaBogotá-Colombia JoséG.Mijares AtopologicalRamseyspaceofinfinitepolyhedraandtherandompolyhedron

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TRS of infinite polyhedra

A topological Ramsey space of infinite polyhedra and the random polyhedron

Jos´e G. Mijares

Pontificia Universidad Javeriana Bogot´a - Colombia

STW 2013

In honour of Ofelia Alas S˜ao Sebasti˜ao, Brasil

August 15, 2013

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TRS of infinite polyhedra Pigeon hole principle

The pigeon hole principle

For every partitionN=C₁∪ · · · ∪ Cr there existsi∈ {1, . . . ,r} such thatC_iis infinite.

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The pigeon hole principle

For every partitionN=C₁∪ · · · ∪ Cr there existsi∈ {1, . . . ,r}

such thatC_iis infinite.

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Ramsey’s theorem 1929

Notation: A^[n]={B⊆A:|B|=n}, A^[<∞]=S

nA^[n]

A^[∞] ={B⊆A:|B|=∞}

Infinite version:For every finite coloring ofN^[2]there is A∈N^[∞]such thatA^[2]is monochromatic.

Finite version:Givenn,r ∈N, there existsM ∈Nsuch that for every coloringc:M^[2]→r, there isA∈M^[n]tal queA^[2]is monochromatic forc.

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Ramsey’s theorem 1929

Notation: A^[n]={B⊆A:|B|=n},

A^[<∞]=S

nA^[n]

A^[∞] ={B⊆A:|B|=∞}

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Ramsey’s theorem 1929

nA^[n]

A^[∞] ={B⊆A:|B|=∞}

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Ramsey’s theorem 1929

nA^[n]

A^[∞]={B⊆A:|B|=∞}

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Ramsey’s theorem 1929

nA^[n]

A^[∞]={B⊆A:|B|=∞}

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Ramsey’s theorem 1929

nA^[n]

A^[∞]={B⊆A:|B|=∞}

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Ramsey’s theorem 1929

nA^[n]

A^[∞]={B⊆A:|B|=∞}

Generalized infinite version:Givenn∈N, for every finite coloring ofN^[n]there isA∈N^[∞]such thatA^[n]is

monochromatic.

Generalized finite version:Givenm,n,r ∈N, there exists M∈Nsuch that for every coloringc:M^[n]→r, there is A∈M^[m]tal queA^[n]is monochromatic forc.

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Ramsey’s theorem 1929

nA^[n]

A^[∞]={B⊆A:|B|=∞}

monochromatic.

Generalized finite version:Givenm,n,r ∈N, there exists M∈Nsuch that for every coloringc:M^[n]→r, there is A∈M^[m]tal queA^[n]is monochromatic forc.

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Ramsey’s theorem 1929

nA^[n]

A^[∞]={B⊆A:|B|=∞}

monochromatic.

Generalized finite version:Givenm,n,r ∈N, there exists M∈Nsuch that for every coloringc:M^[n] →r, there is A∈M^[m]tal queA^[n]is monochromatic forc.

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TRS of infinite polyhedra Ramsey property

Ramsey property

Question:GivenX ⊆N^[∞], is thereA∈N^[∞]such that A^[∞]⊆XorA^[∞]∩X =∅?

Answer:Not in general.

Example:ForA,B∈N^[∞], A∼Biff |A4B|<∞ (AC) Pick an elementB_x of each classx∈N^[∞]/∼, Letcl(A)denote the class ofAand define

X ={A∈N^[∞]:|A4B_cl(A)| is even}

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Ramsey property

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Ramsey property

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Ramsey property

Example:ForA,B∈N^[∞], A∼Biff |A4B|<∞

(AC) Pick an elementB_x of each classx∈N^[∞]/∼, Letcl(A)denote the class ofAand define

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Ramsey property

Example:ForA,B∈N^[∞], A∼Biff |A4B|<∞ (AC) Pick an elementB_x of each classx∈N^[∞]/∼,

Letcl(A)denote the class ofAand define

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Ramsey property

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Ramsey property

[a] ={B∈N^[∞]:a@B} Metric, product topology [a,A] ={B∈N^[∞]:a@B⊆A} Ellentuck’s topology A setX ⊆N^[∞]has theRamsey propertyif for every nonempty[a,A]there isB∈[a,A]such that[a,B]⊆Xor [a,B]∩X =∅.X isRamsey nullif for every nonempty[a,A] there isB∈[a,A]such that[a,B]∩X =∅.

I (AC) There is a set without the Ramsey property.

I (Nash-Williams, 1965) clopen sets are Ramsey.

I (Galvin,1968) Open sets are Ramsey.

I (Galvin-Prikry, 1973) Borel sets are Ramsey.

I (Silver, 1970) Analytic sets are Ramsey.

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Ramsey property

[a] ={B∈N^[∞]:a@B} Metric, product topology

[a,A] ={B∈N^[∞]:a@B⊆A} Ellentuck’s topology A setX ⊆N^[∞]has theRamsey propertyif for every nonempty[a,A]there isB∈[a,A]such that[a,B]⊆Xor [a,B]∩X =∅.X isRamsey nullif for every nonempty[a,A] there isB∈[a,A]such that[a,B]∩X =∅.

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Ramsey property

[a] ={B∈N^[∞]:a@B} Metric, product topology [a,A] ={B∈N^[∞]:a@B⊆A} Ellentuck’s topology

A setX ⊆N^[∞]has theRamsey propertyif for every nonempty[a,A]there isB∈[a,A]such that[a,B]⊆Xor [a,B]∩X =∅.X isRamsey nullif for every nonempty[a,A] there isB∈[a,A]such that[a,B]∩X =∅.

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Ramsey property

[a] ={B∈N^[∞]:a@B} Metric, product topology [a,A] ={B∈N^[∞]:a@B⊆A} Ellentuck’s topology A setX ⊆N^[∞]has theRamsey property

if for every nonempty[a,A]there isB∈[a,A]such that[a,B]⊆Xor [a,B]∩X =∅.X isRamsey nullif for every nonempty[a,A] there isB∈[a,A]such that[a,B]∩X =∅.

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Ramsey property

[a] ={B∈N^[∞]:a@B} Metric, product topology [a,A] ={B∈N^[∞]:a@B⊆A} Ellentuck’s topology A setX ⊆N^[∞]has theRamsey propertyif for every nonempty[a,A]

there isB∈[a,A]such that[a,B]⊆Xor [a,B]∩X =∅.X isRamsey nullif for every nonempty[a,A] there isB∈[a,A]such that[a,B]∩X =∅.

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Ramsey property

[a] ={B∈N^[∞]:a@B} Metric, product topology [a,A] ={B∈N^[∞]:a@B⊆A} Ellentuck’s topology A setX ⊆N^[∞]has theRamsey propertyif for every nonempty[a,A]there isB∈[a,A]such that

[a,B]⊆Xor [a,B]∩X =∅.X isRamsey nullif for every nonempty[a,A] there isB∈[a,A]such that[a,B]∩X =∅.

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Ramsey property

[a] ={B∈N^[∞]:a@B} Metric, product topology [a,A] ={B∈N^[∞]:a@B⊆A} Ellentuck’s topology A setX ⊆N^[∞]has theRamsey propertyif for every nonempty[a,A]there isB∈[a,A]such that[a,B]⊆Xor [a,B]∩X =∅.X isRamsey nullif for every nonempty[a,A]

there isB∈[a,A]such that[a,B]∩X =∅.

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Ramsey property

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Ramsey property

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Ramsey property

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Ramsey property

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Ramsey property

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Ramsey property

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Ellentuck’s theorem – Topological Ramsey theory

[a,A] ={B∈N^[∞]:a@B⊆A} (Ellentuck’s topology) A setX ⊆N^[∞]has theRamsey propertyif for every nonempty[a,A]there isB∈[a,A]such that[a,B]⊆Xor [a,B]∩X =∅.X isRamsey nullif for every nonempty[a,A] there isB∈[a,A]such that[a,B]∩X =∅.

Ellentuck’s theorem, 1974Xis Ramsey iffX has the Baire property (in Ellentuck’s topology).Xis Ramsey null iffXis meager (in Ellentuck’s topology).

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Ellentuck’s theorem – Topological Ramsey theory

[a,A] ={B∈N^[∞]:a@B⊆A} (Ellentuck’s topology)

A setX ⊆N^[∞]has theRamsey propertyif for every nonempty[a,A]there isB∈[a,A]such that[a,B]⊆Xor [a,B]∩X =∅.X isRamsey nullif for every nonempty[a,A] there isB∈[a,A]such that[a,B]∩X =∅.

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Ellentuck’s theorem – Topological Ramsey theory

[a,A] ={B∈N^[∞]:a@B⊆A} (Ellentuck’s topology) A setX ⊆N^[∞]has theRamsey propertyif for every nonempty[a,A]there isB∈[a,A]such that[a,B]⊆Xor [a,B]∩X =∅.X isRamsey nullif for every nonempty[a,A]

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Ellentuck’s theorem – Topological Ramsey theory

[a,A] ={B∈N^[∞]:a@B⊆A} (Ellentuck’s topology) A setX ⊆N^[∞]has theRamsey propertyif for every nonempty[a,A]there isB∈[a,A]such that[a,B]⊆Xor [a,B]∩X =∅.X isRamsey nullif for every nonempty[a,A]

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Topological Ramsey theory

Pigeon hole principle

⇓

Ramsey’s theorem

⇓

Ellentuck’s theorem

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Topological Ramsey theory

⇓

Ramsey’s theorem

⇓

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Topological Ramsey theory

⇓

Ramsey’s theorem

⇓

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Topological Ramsey theory

⇓

Ramsey’s theorem

⇓

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Topological Ramsey theory

Differentpigeon hole principle

⇓

Ramsey-liketheorem

⇓

Ellentuck-liketheorem

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Topological Ramsey theory

⇓

Ramsey-liketheorem

⇓

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Topological Ramsey theory

⇓

Ramsey-liketheorem

⇓

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Topological Ramsey theory

DifferentPHP orDifferentR-like theorem

⇓

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Topological Ramsey theory

DifferentPHP orDifferentR-like theorem

⇓

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Topological Ramsey theory

Some Ellentuck-like theorems

I Ellentuck / Classical PHP.

I Milliken / Hindman’s thm.

I Carlson / Graham-Leeb-Rothschild thm.

I Carlson-Smpson / Dual Ramsey thm.

I Todorcevic / Gowers’ thm.

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Topological Ramsey theory

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Topological Ramsey theory

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Topological Ramsey theory

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Topological Ramsey theory

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Topological Ramsey theory

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TRS of infinite polyhedra Ramsey spaces

Topological Ramsey Spaces – Todorcevic, 2010

(R,≤,r) r:N× R → AR

r_n(A) :=r(n,A) AR_n :={r_n(A) :A∈ R} [a,A] ={B∈ R: (∃n)(a=rn(B)) and (B≤A)}

X ⊆ RisRamseyif for every[a,A]6=∅there isB∈[a,A] such that[a,B]⊆ X or[a,B]∩ X =∅.

(R,≤,r)is atopological Ramsey spaceif subsets ofRwith the Baire property are Ramsey and meager subsets ofRare Ramsey null.

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Topological Ramsey Spaces – Todorcevic, 2010

(R,≤,r) r:N× R → AR

r_n(A) :=r(n,A) AR_n :={r_n(A) :A∈ R} [a,A] ={B∈ R: (∃n)(a=rn(B)) and (B≤A)}

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Topological Ramsey Spaces – Todorcevic, 2010

(R,≤,r) r:N× R → AR r_n(A) :=r(n,A)

AR_n :={r_n(A) :A∈ R} [a,A] ={B∈ R: (∃n)(a=rn(B)) and (B≤A)}

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Topological Ramsey Spaces – Todorcevic, 2010

(R,≤,r) r:N× R → AR

r_n(A) :=r(n,A) AR_n :={r_n(A) :A∈ R}

[a,A] ={B∈ R: (∃n)(a=rn(B)) and (B≤A)}

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Topological Ramsey Spaces – Todorcevic, 2010

(R,≤,r) r:N× R → AR

r_n(A) :=r(n,A) AR_n :={r_n(A) :A∈ R}

[a,A] ={B∈ R: (∃n)(a=rn(B)) and (B≤A)}

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Topological Ramsey Spaces – Todorcevic, 2010

(R,≤,r) r:N× R → AR

r_n(A) :=r(n,A) AR_n :={r_n(A) :A∈ R}

[a,A] ={B∈ R: (∃n)(a=rn(B)) and (B≤A)}

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Topological Ramsey Spaces – Todorcevic, 2010

(R,≤,r) r:N× R → AR

r_n(A) :=r(n,A) AR_n :={r_n(A) :A∈ R}

[a,A] ={B∈ R: (∃n)(a=rn(B)) and (B≤A)}

X ⊆ RisRamseyif for every[a,A]6=∅there isB∈[a,A]

such that[a,B]⊆ X or[a,B]∩ X =∅.

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Topological Ramsey Spaces – Todorcevic, 2010

(R,≤,r) r:N× R → AR

r_n(A) :=r(n,A) AR_n :={r_n(A) :A∈ R}

[a,A] ={B∈ R: (∃n)(a=rn(B)) and (B≤A)}

X ⊆ RisRamseyif for every[a,A]6=∅there isB∈[a,A]

such that[a,B]⊆ X or[a,B]∩ X =∅.

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AXIOMS

(A.1)[Metrization]

(A.1.1) For anyA∈ R,r₀(A) = ∅.

(A.1.2) For anyA,B∈ R, ifA6=Bthen(∃n) (rn(A)6=r_n(B)).

(A.1.3) Ifr_n(A) =r_m(B)thenn=mand(∀i<n) (ri(A) =r_i(B)).

(A.2)[Finitization] There is a quasi order≤finonARsuch that:

(A.2.1) A≤Biff(∀n) (∃m) (rn(A)≤finr_m(B)).

(A.2.2) {b∈ AR:b≤fina}is finite, for everya∈ AR.

(A.2.3) Ifa≤fin bandcvathen there isdvbsuch thatc≤fin d.

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AXIOMS

(A.1)[Metrization]

(A.1.1) For anyA∈ R,r₀(A) = ∅.

(A.1.2) For anyA,B∈ R, ifA6=Bthen(∃n) (rn(A)6=r_n(B)).

(A.1.3) Ifr_n(A) =r_m(B)thenn=mand(∀i<n) (ri(A) =r_i(B)).

(A.2)[Finitization] There is a quasi order≤finonARsuch that:

(A.2.1) A≤Biff(∀n) (∃m) (rn(A)≤finr_m(B)).

(A.2.2) {b∈ AR:b≤fina}is finite, for everya∈ AR.

(A.2.3) Ifa≤fin bandcvathen there isdvbsuch thatc≤fin d.

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AXIOMS

(A.3)[Amalgamation] GivenaandAwith depth_A(a) = n, the following holds:

(i) (∀B∈[n,A]) ([a,B]6=∅).

(ii) (∀B∈[a,A]) (∃A⁰ ∈[n,A]) ([a,A⁰]⊆[a,B]).

(A.4)[Pigeonhole Principle] For everyA∈ Rand every O ⊆ AR₁there isB≤Asuch thatAR₁|B⊆ Oor AR₁|B⊆ O^c.

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AXIOMS

(A.3)[Amalgamation] GivenaandAwith depth_A(a) = n, the following holds:

(i) (∀B∈[n,A]) ([a,B]6=∅).

(ii) (∀B∈[a,A]) (∃A⁰ ∈[n,A]) ([a,A⁰]⊆[a,B]).

(A.4)[Pigeonhole Principle] For everyA∈ Rand every O ⊆ AR₁there isB≤Asuch thatAR₁|B⊆ Oor AR₁|B⊆ O^c.

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Abstract Ellentuck theorem

Abstract Ellentuck theorem.Any(R,≤,r)withRmetrically closed and satisfying (A.1)-(A.4) is a topological Ramsey space.

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A topological Ramsey space of infinite polyhedra

TRS of of infinite polyhedra, Mijares - Padilla, 2012/2013

Elements:LetP be the collection of pairs(X,S_X)such that:

1. X∈N^[∞],

2. S_X ⊆X^[<∞]ishereditary, i.e.,u⊆v&v∈S_X ⇒u∈S_X, and

3. S

SX =S

{u:u∈SX}=X.

Pre-ordering: (Y,S_Y)≤(X,S_X) ⇔ Y ⊆X&S_Y ⊆S_X. Approximations:r(n,(X,S_X)) =r_n(X,S_X) = (X n,S_X n). Theorem:(M-Padilla, 2013+)(P,≤,r)is a TRS.

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TRS of of infinite polyhedra, Mijares - Padilla, 2012/2013

1. X∈N^[∞],

3. S

SX =S

{u:u∈SX}=X.

Pre-ordering: (Y,S_Y)≤(X,S_X) ⇔ Y ⊆X&S_Y ⊆S_X.

Approximations:r(n,(X,S_X)) =r_n(X,S_X) = (X n,S_X n). Theorem:(M-Padilla, 2013+)(P,≤,r)is a TRS.

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TRS of of infinite polyhedra, Mijares - Padilla, 2012/2013

1. X∈N^[∞],

3. S

SX =S

{u:u∈SX}=X.

Pre-ordering: (Y,S_Y)≤(X,S_X) ⇔ Y ⊆X&S_Y ⊆S_X. Approximations:r(n,(X,S_X)) =r_n(X,S_X) = (X n,S_X n).

Theorem:(M-Padilla, 2013+)(P,≤,r)is a TRS.

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TRS of of infinite polyhedra, Mijares - Padilla, 2012/2013

1. X∈N^[∞],

3. S

SX =S

{u:u∈SX}=X.

Theorem:(M-Padilla, 2013+)(P,≤,r)is a TRS.

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Subspaces, Mijares - Padilla, 2012/2013

Elements:Givenk ∈N, letP(k)be the collection of pairs (X,S_X)such that:

1. X∈N^[∞],

2. S_X ⊆X^[≤k]ishereditary, i.e.,u⊆v&v∈S_X ⇒u∈S_X, and

3. S

SX =S

{u:u∈SX}=X.

Pre-ordering: (Y,S_Y)≤(X,S_X) ⇔ Y ⊆X&S_Y ⊆S_X. Approximations:r(n,(X,S_X)) =r_n(X,S_X) = (X n,S_X n). Theorem:(M-Padilla, 2013+) For eachk,(P(k),≤,r)is a TRS.

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Subspaces, Mijares - Padilla, 2012/2013

1. X∈N^[∞],

3. S

SX =S

{u:u∈SX}=X.

Pre-ordering: (Y,S_Y)≤(X,S_X) ⇔ Y ⊆X&S_Y ⊆S_X.

Approximations:r(n,(X,S_X)) =r_n(X,S_X) = (X n,S_X n). Theorem:(M-Padilla, 2013+) For eachk,(P(k),≤,r)is a TRS.

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Subspaces, Mijares - Padilla, 2012/2013

1. X∈N^[∞],

3. S

SX =S

{u:u∈SX}=X.

Theorem:(M-Padilla, 2013+) For eachk,(P(k),≤,r)is a TRS.

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Subspaces, Mijares - Padilla, 2012/2013

1. X∈N^[∞],

3. S

SX =S

{u:u∈SX}=X.

Theorem:(M-Padilla, 2013+) For eachk,(P(k),≤,r)is a TRS.

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TRS of infinite polyhedra Ramsey classes

Ramsey classes

Asignature:L=<(Ri)i∈I,(Fj)j∈J > (Ri)i∈I is a setrelation symbols (Fj)j∈J is a set offunction symbols

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Ramsey classes

Asignature:L=<(Ri)i∈I,(Fj)j∈J >

(Ri)i∈I is a setrelation symbols (Fj)j∈J is a set offunction symbols

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Ramsey classes

(Ri)i∈I is a setrelation symbols

(Fj)j∈J is a set offunction symbols

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Ramsey classes

(Ri)i∈I is a setrelation symbols (Fj)j∈J is a set offunction symbols

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Structures

L-structure,A=<A,(R^A_i )i∈I,(F^A_j )j∈J >:

•A non empty setA6=∅called theuniverseof the structure;

•a set of relations(R^A_i )i∈I whereR^A_i ⊆Aⁿ⁽ⁱ⁾for eachi∈I; and

•a set of functions(F_j^A)j∈J whereF_j^A:A^m(j) -Afor each j∈J.

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Structures

AmorphismofL-structuresA ^π^-Bis a mapA ^π-B

• a₁, . . . ,a_n(i)

∈R^A_i iff π(a₁), . . . , π a_n(i)

∈R^B_i; for all a₁, . . . ,a_n(i)∈A.

•π F^A_j a₁, . . .a_m(j)

=F^B_j π(a₁), . . . , π a_m(j) for all a₁, . . . ,a_m(j) ∈A.

Whenπis injective we say that it is anembedding. In particular, we say thatAis asubstructureofB, and write A≤BwheneverA⊆Band the inclusion map is an embedding.

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Fra¨ıss´e class of finite structure

A non empty class of finite relationalL-structuresC is a Fra¨ıss´e classif it satisfies the following:

1. C is closed under isomorphisms: IfA∈ C andA∼=Bthen B∈ C.

2. C is hereditary: IfA∈ C andB≤AthenB∈ C.

3. C contains structures with arbitrarily high finite cardinality. 4. Joint embedding property: IfA,B∈ C then there isD∈ C

such thatA≤DandB≤D.

5. Amalgamation property: GivenA,B1,B2 ∈ Cand embeddingsA ^fⁱ ^-Bⁱ,i∈ {1,2}, there isD∈ C and embeddingsBⁱ ^gⁱ^-Dsuch thatg₁◦f₁ =g₂◦f₂.

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Fra¨ıss´e class of finite structure

2. C is hereditary: IfA∈ C andB≤AthenB∈ C.

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Fra¨ıss´e class of finite structure

2. C is hereditary: IfA∈ CandB≤AthenB∈ C.

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Fra¨ıss´e class of finite structure

3. C contains structures with arbitrarily high finite cardinality.

4. Joint embedding property: IfA,B∈ C then there isD∈ C such thatA≤DandB≤D.

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Fra¨ıss´e class of finite structure

4. Joint embedding property: IfA,B∈ Cthen there isD∈ C such thatA≤DandB≤D.

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Fra¨ıss´e class of finite structure

4. Joint embedding property: IfA,B∈ Cthen there isD∈ C such thatA≤DandB≤D.

5. Amalgamation property: GivenA,B1,B2∈ C and embeddingsA ^fⁱ ^-Bⁱ,i∈ {1,2}, there isD∈ C and embeddingsBⁱ ^gⁱ^-Dsuch thatg₁◦f₁ =g₂◦f₂.

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Fra¨ıss´e limit

IIfC is Fra¨ıss´e , then there is a unique (up to isomorphism) countably infiniteultrahomogeneousstructureFsuch that

C = the class of finite substructures of F

=:Age(F).

ThisFis theFra¨ıss´e limitofC and we writeF=FLim(C).

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Fra¨ıss´e limit

C = the class of finite substructures of F=:Age(F).

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Fra¨ıss´e limit

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Fra¨ıss´e limit

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Ramsey class of finite structures

A Fra¨ıss´e classC has theRamsey propertyiff, for every integerr>1 and everyA,B∈ C such thatA≤B, there is C∈ C such that for eachr-coloring

c: C

A

-r

of the set C

A

, there existsB⁰ ∈ C

B

such that B⁰

A

is monochromatic.

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Ramsey class of finite structures

A Fra¨ıss´e classC has theRamsey propertyiff, for every integerr>1 and everyA,B∈ C such thatA≤B,

there is C∈ C such that for eachr-coloring

c: C

A

-r

of the set C

A

B

such that B⁰

A

is monochromatic.

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Ramsey class of finite structures

A Fra¨ıss´e classC has theRamsey propertyiff, for every integerr>1 and everyA,B∈ C such thatA≤B, there is C∈ C such that

for eachr-coloring

c: C

A

-r

of the set C

A

B

such that B⁰

A

is monochromatic.

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Ramsey class of finite structures

c: C

A

-r

of the set C

A

,

there existsB⁰ ∈ C

B

such that B⁰

A

is monochromatic.

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Ramsey class of finite structures

c: C

A

-r

of the set C

A

B

such that B⁰

A

is monochromatic.

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Extreme amenability

A topological groupGisextremely amenableor hasthe fixed point on compacta property, if for every continuous action of Gon a compact spaceX there existsx∈Xsuch that for every g∈G,g·x=x.

Theorem

(Pestov 2006) LetFbe a countably infinite ultrahomogeneous structure andC =Age(F). The polish groupAut(F)is

extremely amenable if and only ifC has the Ramsey property and all the structures ofC are rigid.

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Extreme amenability

Theorem

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Extreme amenability

Theorem

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Finite polyhedra as a Ramsey class

ConsiderL=<(R_i)_i∈_N_\{0} >, a signature with an infinite number of relational symbols such that for eachi∈Nthe arity ofR_iisn(i) =i.

LetKP the class of finite ordered polyhedra.KP is a class of L∪ {<}-structures.

Facts:

I AP ⊆ KP.

I For everyA∈ KP there is(a,S_a)∈ APsuch that A∼= (a,S_a). Actually,KP is the closure ofAP under isomorphisms.

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Finite polyhedra as a Ramsey class

LetKP the class of finite ordered polyhedra.

KP is a class of L∪ {<}-structures.

Facts:

I AP ⊆ KP.

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Finite polyhedra as a Ramsey class

Facts:

I AP ⊆ KP.

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Finite polyhedra as a Ramsey class

Facts:

I AP ⊆ KP.

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Finite polyhedra as a Ramsey class

Facts:

I AP ⊆ KP.

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Finite polyhedra as a Ramsey class

Theorem

The classKP of all finite ordered polyhedra is Ramsey.

Corollary

LetP=FLim(KP), the Fra¨ıss´e limit ofKP. Then,Aut(P)with the Polish topology inherited from S_∞is extremely amenable.

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TRS of infinite polyhedra The random polyhedron

The random polyhedron

Consider a countably infinite setω. Define a family S_ω ⊆ω^[<∞], as follows:

Hold a coin. Define a familyT_ω ⊆ω^[<∞]probabilistically in the following way: for everyu∈ω^[<∞]such that|u|>1 flip the coin, and say thatuis inT_ωif and only if you get heads. Set

S_ω :=ω^[1]∪ {v: (∃u∈T_ω)v⊆u} (1) (ω,S_ω)is an infiniterandompolyhedron.

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The random polyhedron

Consider a countably infinite setω.

Define a family S_ω ⊆ω^[<∞], as follows:

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The random polyhedron

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The random polyhedron

Hold a coin.

Define a familyT_ω ⊆ω^[<∞]probabilistically in the following way: for everyu∈ω^[<∞]such that|u|>1 flip the coin, and say thatuis inT_ωif and only if you get heads. Set

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The random polyhedron

Hold a coin. Define a familyT_ω ⊆ω^[<∞]probabilistically in the following way: for everyu∈ω^[<∞]such that|u|>1 flip the coin, and say thatuis inT_ωif and only if you get heads.

Set