[PPT] - Selective for R but not Ramsey for R Timothy Onofre Trujillo PowerPoint Presentation

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Selective for R but not Ramsey for R

Timothy Onofre Trujillo

University of Denver

BLAST 2013 – Chapman University August 9, 2013

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 1 / 19

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Outline

1

Background Notation Selective ultrafilters on ω

2

Topological Ramsey Theory Definition of a topological Ramsey space The topological Ramsey space R1. The topological Ramsey space R⋆

3

Selective but not Ramsey ultrafilters R1 Rn

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 2 / 19

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Background Notation

Definition For each S ⊆ ω<ω, [S] = {s ∈ S : ∀t ∈ S, s ⊑ t ⇒ s = t} cl(S) = {t ∈ ω<ω : t ⊑ s ∈ S} π0(S) = {s0 : s ∈ S} S is a Tree on ω, if cl(S) = S. For s, t ∈ ω<ω, s ≤ t ⇔ (s ⊑ t or |s| = |t| & s ≤lex t) If S and T are trees on ω then T S

= {U ⊆ T : U ∼

= S}.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 3 / 19

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Background Selective ultrafilters on ω

Definition Let U be an ultrafilter on ω.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 4 / 19

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Background Selective ultrafilters on ω

Definition Let U be an ultrafilter on ω.

1 U is selective if for each sequence A0 ⊇ A1 ⊇ A2 ⊇ . . . of members

f U, there exists A = {a0, a1, . . . } ∈ U such that for each n < ω,

A \ {a0, a1, . . . , an−1} ⊆ An.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 4 / 19

SLIDE 6

Background Selective ultrafilters on ω

Definition Let U be an ultrafilter on ω.

1 U is selective if for each sequence A0 ⊇ A1 ⊇ A2 ⊇ . . . of members

f U, there exists A = {a0, a1, . . . } ∈ U such that for each n < ω,

A \ {a0, a1, . . . , an−1} ⊆ An.

2 U is Ramsey if for each map F : [ω]n → 2 there exists A ∈ U such

that F is constant on [A]n

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 4 / 19

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Background Selective ultrafilters on ω

Definition Let U be an ultrafilter on ω.

1 U is selective if for each sequence A0 ⊇ A1 ⊇ A2 ⊇ . . . of members

f U, there exists A = {a0, a1, . . . } ∈ U such that for each n < ω,

A \ {a0, a1, . . . , an−1} ⊆ An.

2 U is Ramsey if for each map F : [ω]n → 2 there exists A ∈ U such

that F is constant on [A]n Theorem (Kunen, [1]) Let U be an ultrafilter on ω. U is selective if and only if U is Ramsey.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 4 / 19

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Topological Ramsey Theory Definition of a topological Ramsey space

Definition ([6]) Let (R, ≤, r) be a triple,

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 5 / 19

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Topological Ramsey Theory Definition of a topological Ramsey space

Definition ([6]) Let (R, ≤, r) be a triple, where R is nonempty,

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 5 / 19

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Topological Ramsey Theory Definition of a topological Ramsey space

Definition ([6]) Let (R, ≤, r) be a triple, where R is nonempty, where ≤ is a quasi-ordering on R and

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 5 / 19

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Topological Ramsey Theory Definition of a topological Ramsey space

Definition ([6]) Let (R, ≤, r) be a triple, where R is nonempty, where ≤ is a quasi-ordering on R and where r : R × ω → AR is a mapping giving us the sequence (rn(·) = r(·, n)) of approximation mappings.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 5 / 19

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Topological Ramsey Theory Definition of a topological Ramsey space

Definition ([6]) Let (R, ≤, r) be a triple, where R is nonempty, where ≤ is a quasi-ordering on R and where r : R × ω → AR is a mapping giving us the sequence (rn(·) = r(·, n)) of approximation mappings. For s ∈ AR and X ∈ R let [s, X] = {Y ∈ R : Y ≤ X & (∃n) s = rn(Y )}.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 5 / 19

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Topological Ramsey Theory Definition of a topological Ramsey space

Definition ([6]) Let (R, ≤, r) be a triple, where R is nonempty, where ≤ is a quasi-ordering on R and where r : R × ω → AR is a mapping giving us the sequence (rn(·) = r(·, n)) of approximation mappings. For s ∈ AR and X ∈ R let [s, X] = {Y ∈ R : Y ≤ X & (∃n) s = rn(Y )}. The Ellentuck topology on R is the topology generated by the collection {[s, X] : s ∈ AR, X ∈ R} .

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 5 / 19

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Topological Ramsey Theory Definition of a topological Ramsey space

Definition ([6]) Let (R, ≤, r) be a triple, where R is nonempty, where ≤ is a quasi-ordering on R and where r : R × ω → AR is a mapping giving us the sequence (rn(·) = r(·, n)) of approximation mappings. For s ∈ AR and X ∈ R let [s, X] = {Y ∈ R : Y ≤ X & (∃n) s = rn(Y )}. The Ellentuck topology on R is the topology generated by the collection {[s, X] : s ∈ AR, X ∈ R} . Example (The Ellentuck Space, ([ω]ω, ⊆, r)) rn({a0, a1, a2, . . . }) = {a0, . . . , an−1}

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 5 / 19

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Topological Ramsey Theory Definition of a topological Ramsey space

Definition ([6]) A subset X of R is Ramsey if for every nonempty [s, X], there is a Y ∈ [s, X] such that [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 6 / 19

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Topological Ramsey Theory Definition of a topological Ramsey space

Definition ([6]) A subset X of R is Ramsey if for every nonempty [s, X], there is a Y ∈ [s, X] such that [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅. X is Ramsey null if for every nonempty [s, X], there exists Y ∈ [s, X] such that [s, Y ] ∩ X = ∅.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 6 / 19

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Topological Ramsey Theory Definition of a topological Ramsey space

Definition ([6]) A subset X of R is Ramsey if for every nonempty [s, X], there is a Y ∈ [s, X] such that [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅. X is Ramsey null if for every nonempty [s, X], there exists Y ∈ [s, X] such that [s, Y ] ∩ X = ∅. A triple (R, ≤, r) is a topological Ramsey space if every subset of R with the Baire property is Ramsey and if every meager subset of R is Ramsey null.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 6 / 19

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Topological Ramsey Theory Definition of a topological Ramsey space

Definition ([6]) A subset X of R is Ramsey if for every nonempty [s, X], there is a Y ∈ [s, X] such that [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅. X is Ramsey null if for every nonempty [s, X], there exists Y ∈ [s, X] such that [s, Y ] ∩ X = ∅. A triple (R, ≤, r) is a topological Ramsey space if every subset of R with the Baire property is Ramsey and if every meager subset of R is Ramsey null. The Ellentuck Theorem (Ellentuck, [3]) The Ellentuck space ([ω]ω, ⊆, r) is a topological Ramsey space.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 6 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition For each n < ω, let T1(n) = { , n , n, i : i ≤ n}.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 7 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition For each n < ω, let T1(n) = { , n , n, i : i ≤ n}.

,
T1(0)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 7 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition For each n < ω, let T1(n) = { , n , n, i : i ≤ n}.

1

,

1

, 1

1

T1(1)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 7 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition For each n < ω, let T1(n) = { , n , n, i : i ≤ n}.

,
1

,

1

, 1

1
2

,

2

, 1

2

, 2

2
3

,

3

, 1

3

, 2

3

, 3

3
4

,

4

, 1

4

, 2

4

, 3

4

, 4

4
5

,

5

, 1

5

, 2

5

, 3

5

, 4

5

, 5

5

· · ·

T1 =

n<ω

T1(n)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 7 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition R1 = T1 T1

.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 8 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition R1 = T1 T1

.

For each S ∈ R1 and each i < ω

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 8 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition R1 = T1 T1

.

For each S ∈ R1 and each i < ω , let π0(S) = {k0, k1, k2, . . . }

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 8 / 19

SLIDE 26

Topological Ramsey Theory The topological Ramsey space R1.

Definition R1 = T1 T1

.

For each S ∈ R1 and each i < ω , let π0(S) = {k0, k1, k2, . . . } S(i) = {s ∈ S : π0(s) = ki} ∪ { }

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 8 / 19

SLIDE 27

Topological Ramsey Theory The topological Ramsey space R1.

Definition R1 = T1 T1

.

For each S ∈ R1 and each i < ω , let π0(S) = {k0, k1, k2, . . . } S(i) = {s ∈ S : π0(s) = ki} ∪ { } ri(S) =

j<i

S(j).

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 8 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition R1 = T1 T1

.

For each S ∈ R1 and each i < ω , let π0(S) = {k0, k1, k2, . . . } S(i) = {s ∈ S : π0(s) = ki} ∪ { } ri(S) =

j<i

S(j). For S, T ∈ R1, S ≤ T if and only if S is a subtree of T.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 8 / 19

SLIDE 29

Topological Ramsey Theory The topological Ramsey space R1.

Definition R1 = T1 T1

.

For each S ∈ R1 and each i < ω , let π0(S) = {k0, k1, k2, . . . } S(i) = {s ∈ S : π0(s) = ki} ∪ { } ri(S) =

j<i

S(j). For S, T ∈ R1, S ≤ T if and only if S is a subtree of T. Theorem (Dobrinen, Todorcevic, [2]) (R1, ≤, r) is a topological Ramsey space.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 8 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition Let U be an ultrafilter on [T1] and C ⊆ R1.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 9 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition Let U be an ultrafilter on [T1] and C ⊆ R1.

1 U is generated by C, if

{[A] : A ∈ C} is cofinal in (U, ⊇).

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 9 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition Let U be an ultrafilter on [T1] and C ⊆ R1.

1 U is generated by C, if

{[A] : A ∈ C} is cofinal in (U, ⊇).

2 U is selective for R1, if for each sequence A0 ⊇ A1 ⊇ A2 ⊇ . . . of

members of C, there exists A ∈ C such that for each n < ω, A \ rn(A) ⊆ An.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 9 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition Let U be an ultrafilter on [T1] and C ⊆ R1.

1 U is generated by C, if

{[A] : A ∈ C} is cofinal in (U, ⊇).

2 U is selective for R1, if for each sequence A0 ⊇ A1 ⊇ A2 ⊇ . . . of

members of C, there exists A ∈ C such that for each n < ω, A \ rn(A) ⊆ An.

3 U is Ramsey for R1, if for map F : ARn → 2 there exists A ∈ C

such that F is constant on ARn|A = {rn(B) : B ≤ A}.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 9 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition For S, T ∈ R1, S ≤∗ T ⇐ ⇒ (∃i < ω)(S \ ri(S) ⊆ T).

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 10 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition For S, T ∈ R1, S ≤∗ T ⇐ ⇒ (∃i < ω)(S \ ri(S) ⊆ T). Theorem (Mijares,[5]) If C is a (R1, ≤∗)-generic filter then C generates a Ramsey for R1 ultrafilter on [T1].

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 10 / 19

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Topological Ramsey Theory The topological Ramsey space R1.

Definition For S, T ∈ R1, S ≤∗ T ⇐ ⇒ (∃i < ω)(S \ ri(S) ⊆ T). Theorem (Mijares,[5]) If C is a (R1, ≤∗)-generic filter then C generates a Ramsey for R1 ultrafilter on [T1]. Theorem (Mijares,[5]) If U is Ramsey for R1 then U is selective for R1.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 10 / 19

SLIDE 37

Topological Ramsey Theory The topological Ramsey space R1.

Definition For S, T ∈ R1, S ≤∗ T ⇐ ⇒ (∃i < ω)(S \ ri(S) ⊆ T). Theorem (Mijares,[5]) If C is a (R1, ≤∗)-generic filter then C generates a Ramsey for R1 ultrafilter on [T1]. Theorem (Mijares,[5]) If U is Ramsey for R1 then U is selective for R1. Question Is Ramsey for R1 equivalent to selective for R1?

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 10 / 19

SLIDE 38

Topological Ramsey Theory The topological Ramsey space R1.

Definition For S, T ∈ R1, S ≤∗ T ⇐ ⇒ (∃i < ω)(S \ ri(S) ⊆ T). Theorem (Mijares,[5]) If C is a (R1, ≤∗)-generic filter then C generates a Ramsey for R1 ultrafilter on [T1]. Theorem (Mijares,[5]) If U is Ramsey for R1 then U is selective for R1. Question Is Ramsey for R1 equivalent to selective for R1? Lemma (Follows from work of Laflamme, [4]) If U is Ramsey for R1 then U is weakly-Ramsey.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 10 / 19

SLIDE 39

Topological Ramsey Theory The topological Ramsey space R⋆

Definition Let {t0, t1, t2, . . . } be an increasing enumeration of [T1].

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 11 / 19

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Topological Ramsey Theory The topological Ramsey space R⋆

Definition Let {t0, t1, t2, . . . } be an increasing enumeration of [T1]. For each n < ω, let T ⋆

1 (n) = cl ({n⌢ ti : i ≤ n}) .

T ⋆

1 =

n<ω

T ⋆

1 (n)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 11 / 19

SLIDE 41

Topological Ramsey Theory The topological Ramsey space R⋆

Definition Let {t0, t1, t2, . . . } be an increasing enumeration of [T1]. For each n < ω, let T ⋆

1 (n) = cl ({n⌢ ti : i ≤ n}) .

⌢

t

T ⋆

1 =

n<ω

T ⋆

1 (n)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 11 / 19

SLIDE 42

Topological Ramsey Theory The topological Ramsey space R⋆

Definition Let {t0, t1, t2, . . . } be an increasing enumeration of [T1]. For each n < ω, let T ⋆

1 (n) = cl ({n⌢ ti : i ≤ n}) .

⌢

t

1
⌢

t

1
⌢

t

1

T ⋆

1 =

n<ω

T ⋆

1 (n)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 11 / 19

SLIDE 43

Topological Ramsey Theory The topological Ramsey space R⋆

Definition Let {t0, t1, t2, . . . } be an increasing enumeration of [T1]. For each n < ω, let T ⋆

1 (n) = cl ({n⌢ ti : i ≤ n}) .

⌢

t

1
⌢

t

1
⌢

t

1

2
⌢

t

2
⌢

t

1

2
⌢

t

2

T ⋆

1 =

n<ω

T ⋆

1 (n)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 11 / 19

SLIDE 44

Topological Ramsey Theory The topological Ramsey space R⋆

Definition Let {t0, t1, t2, . . . } be an increasing enumeration of [T1]. For each n < ω, let T ⋆

1 (n) = cl ({n⌢ ti : i ≤ n}) .

⌢

t

1
⌢

t

1
⌢

t

1

2
⌢

t

2
⌢

t

1

2
⌢

t

2

3
⌢

t

3
⌢

t

1

3
⌢

t

2

3
⌢

t

3

T ⋆

1 =

n<ω

T ⋆

1 (n)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 11 / 19

SLIDE 45

Topological Ramsey Theory The topological Ramsey space R⋆

Definition Let {t0, t1, t2, . . . } be an increasing enumeration of [T1]. For each n < ω, let T ⋆

1 (n) = cl ({n⌢ ti : i ≤ n}) .

⌢

t

1
⌢

t

1
⌢

t

1

2
⌢

t

2
⌢

t

1

2
⌢

t

2

3
⌢

t

3
⌢

t

1

3
⌢

t

2

3
⌢

t

3

4
⌢

t

4
⌢

t

1

4
⌢

t

2

4
⌢

t

3

4
⌢

t

4

T ⋆

1 =

n<ω

T ⋆

1 (n)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 11 / 19

SLIDE 46

Topological Ramsey Theory The topological Ramsey space R⋆

Definition Let {t0, t1, t2, . . . } be an increasing enumeration of [T1]. For each n < ω, let T ⋆

1 (n) = cl ({n⌢ ti : i ≤ n}) .

⌢

t

1
⌢

t

1
⌢

t

1

2
⌢

t

2
⌢

t

1

2
⌢

t

2

3
⌢

t

3
⌢

t

1

3
⌢

t

2

3
⌢

t

3

4
⌢

t

4
⌢

t

1

4
⌢

t

2

4
⌢

t

3

4
⌢

t

4

5
⌢

t

5
⌢

t

1

5
⌢

t

2

5
⌢

t

3

5
⌢

t

4

5
⌢

t

5

· · ·

T ⋆

1 =

n<ω

T ⋆

1 (n)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 11 / 19

SLIDE 47

Topological Ramsey Theory The topological Ramsey space R⋆

Definition R⋆

1 =

T ⋆

1

T ⋆

1

.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 12 / 19

SLIDE 48

Topological Ramsey Theory The topological Ramsey space R⋆

Definition R⋆

1 =

T ⋆

1

T ⋆

1

.

For each S ∈ R⋆

1 and each i < ω

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 12 / 19

SLIDE 49

Topological Ramsey Theory The topological Ramsey space R⋆

Definition R⋆

1 =

T ⋆

1

T ⋆

1

.

For each S ∈ R⋆

1 and each i < ω , let

π0(S) = {k0, k1, k2, . . . }

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 12 / 19

SLIDE 50

Topological Ramsey Theory The topological Ramsey space R⋆

Definition R⋆

1 =

T ⋆

1

T ⋆

1

.

For each S ∈ R⋆

1 and each i < ω , let

π0(S) = {k0, k1, k2, . . . } S(i) = {s ∈ S : π0(s) = ki} ∪ { }

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 12 / 19

SLIDE 51

Topological Ramsey Theory The topological Ramsey space R⋆

Definition R⋆

1 =

T ⋆

1

T ⋆

1

.

For each S ∈ R⋆

1 and each i < ω , let

π0(S) = {k0, k1, k2, . . . } S(i) = {s ∈ S : π0(s) = ki} ∪ { } ri(S) =

j<i

S(j).

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 12 / 19

SLIDE 52

Topological Ramsey Theory The topological Ramsey space R⋆

Definition R⋆

1 =

T ⋆

1

T ⋆

1

.

For each S ∈ R⋆

1 and each i < ω , let

π0(S) = {k0, k1, k2, . . . } S(i) = {s ∈ S : π0(s) = ki} ∪ { } ri(S) =

j<i

S(j). For S, T ∈ R⋆

1,

S ≤ T if and only if S is a subtree of T.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 12 / 19

SLIDE 53

Topological Ramsey Theory The topological Ramsey space R⋆

Definition R⋆

1 =

T ⋆

1

T ⋆

1

.

For each S ∈ R⋆

1 and each i < ω , let

π0(S) = {k0, k1, k2, . . . } S(i) = {s ∈ S : π0(s) = ki} ∪ { } ri(S) =

j<i

S(j). For S, T ∈ R⋆

1,

S ≤ T if and only if S is a subtree of T. Theorem (T.) (R⋆

1, ≤, r) is a topological Ramsey space.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 12 / 19

SLIDE 54

Topological Ramsey Theory The topological Ramsey space R⋆

Theorem (Mijares, [5]) If C is a (R⋆

1, ≤∗)-generic filter then C generates a Ramsey for R⋆ 1

ultrafilter on [T ⋆

1 ].

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 13 / 19

SLIDE 55

Topological Ramsey Theory The topological Ramsey space R⋆

Theorem (Mijares, [5]) If C is a (R⋆

1, ≤∗)-generic filter then C generates a Ramsey for R⋆ 1

ultrafilter on [T ⋆

1 ].

Definition Let {s0, s1, s2, . . . } be the increasing enumeration of [T ⋆

1 ].

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 13 / 19

SLIDE 56

Topological Ramsey Theory The topological Ramsey space R⋆

Theorem (Mijares, [5]) If C is a (R⋆

1, ≤∗)-generic filter then C generates a Ramsey for R⋆ 1

ultrafilter on [T ⋆

1 ].

Definition Let {s0, s1, s2, . . . } be the increasing enumeration of [T ⋆

1 ].

δ : [T ⋆

1 ] → [T1] and Γ : R⋆ 1 → R1

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 13 / 19

SLIDE 57

Topological Ramsey Theory The topological Ramsey space R⋆

Theorem (Mijares, [5]) If C is a (R⋆

1, ≤∗)-generic filter then C generates a Ramsey for R⋆ 1

ultrafilter on [T ⋆

1 ].

Definition Let {s0, s1, s2, . . . } be the increasing enumeration of [T ⋆

1 ].

δ : [T ⋆

1 ] → [T1] and Γ : R⋆ 1 → R1

δ(sj) = tj and Γ(S) = cl(δ′′[S])

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 13 / 19

SLIDE 58

Topological Ramsey Theory The topological Ramsey space R⋆

1 2 3 4 5

· · ·

1 2 3 4 5

· · ·

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 14 / 19

SLIDE 59

Topological Ramsey Theory The topological Ramsey space R⋆

1 2 3 4 5

· · ·

1 2 3 4 5

· · ·

1 2 4 5

· · ·

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 14 / 19

SLIDE 60

Topological Ramsey Theory The topological Ramsey space R⋆

1 2 3 4 5

· · ·

1 2 3 4 5

· · · · · ·

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 14 / 19

SLIDE 61

Topological Ramsey Theory The topological Ramsey space R⋆

1 2 3 4 5

· · ·

1 2 3 4 5

· · · · · ·

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 14 / 19

SLIDE 62

Topological Ramsey Theory The topological Ramsey space R⋆

1 2 3 4 5

· · ·

1 2 3 4 5

· · · · · ·

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 14 / 19

SLIDE 63

Selective but not Ramsey ultrafilters R1

Theorem (T., [7]) If C is a (R⋆

1, ≤∗)-generic filter then Γ′′C generates an ultrafilter on [T1]

which is selective for R1 but not Ramsey for R1.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 15 / 19

SLIDE 64

Selective but not Ramsey ultrafilters R1

Theorem (T., [7]) If C is a (R⋆

1, ≤∗)-generic filter then Γ′′C generates an ultrafilter on [T1]

which is selective for R1 but not Ramsey for R1. Proof. Let U be the ultrafilter on [T ⋆

1 ] generated by C.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 15 / 19

SLIDE 65

Selective but not Ramsey ultrafilters R1

Theorem (T., [7]) If C is a (R⋆

1, ≤∗)-generic filter then Γ′′C generates an ultrafilter on [T1]

which is selective for R1 but not Ramsey for R1. Proof. Let U be the ultrafilter on [T ⋆

1 ] generated by C. δ(U) is an ultrafilter on

[T1] generated by Γ′′C.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 15 / 19

SLIDE 66

Selective but not Ramsey ultrafilters R1

Theorem (T., [7]) If C is a (R⋆

1, ≤∗)-generic filter then Γ′′C generates an ultrafilter on [T1]

which is selective for R1 but not Ramsey for R1. Proof. Let U be the ultrafilter on [T ⋆

1 ] generated by C. δ(U) is an ultrafilter on

[T1] generated by Γ′′C. Γ(A0) ⊇ Γ(A1) ⊇ Γ(A2) ⊇ . . . A0 ⊇ A1 ⊇ A2 ⊇ . . . ∃A ∈ C, ∀i < ω, A \ ri(A) ⊆ Ai ∀i < ω, Γ(A) \ ri(Γ(A)) ⊆ Γ(Ai)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 15 / 19

SLIDE 67

Selective but not Ramsey ultrafilters R1

Theorem (T., [7]) If C is a (R⋆

1, ≤∗)-generic filter then Γ′′C generates an ultrafilter on [T1]

which is selective for R1 but not Ramsey for R1. Proof. Let U be the ultrafilter on [T ⋆

1 ] generated by C. δ(U) is an ultrafilter on

[T1] generated by Γ′′C. Γ(A0) ⊇ Γ(A1) ⊇ Γ(A2) ⊇ . . . A0 ⊇ A1 ⊇ A2 ⊇ . . . ∃A ∈ C, ∀i < ω, A \ ri(A) ⊆ Ai ∀i < ω, Γ(A) \ ri(Γ(A)) ⊆ Γ(Ai) Let F : [T1]2 → 3 be the map such that F{s, t} is the length of the longest common initial segment of δ−1(s) and δ−1(t).

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 15 / 19

SLIDE 68

Selective but not Ramsey ultrafilters R1

Theorem (T., [7]) If C is a (R⋆

1, ≤∗)-generic filter then Γ′′C generates an ultrafilter on [T1]

which is selective for R1 but not Ramsey for R1. Proof. Let U be the ultrafilter on [T ⋆

1 ] generated by C. δ(U) is an ultrafilter on

[T1] generated by Γ′′C. Γ(A0) ⊇ Γ(A1) ⊇ Γ(A2) ⊇ . . . A0 ⊇ A1 ⊇ A2 ⊇ . . . ∃A ∈ C, ∀i < ω, A \ ri(A) ⊆ Ai ∀i < ω, Γ(A) \ ri(Γ(A)) ⊆ Γ(Ai) Let F : [T1]2 → 3 be the map such that F{s, t} is the length of the longest common initial segment of δ−1(s) and δ−1(t). For each A ∈ C, F does not

mit a value on [Γ(A)]2.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 15 / 19

SLIDE 69

Selective but not Ramsey ultrafilters R1

Theorem (T., [7]) If C is a (R⋆

1, ≤∗)-generic filter then Γ′′C generates an ultrafilter on [T1]

which is selective for R1 but not Ramsey for R1. Proof. Let U be the ultrafilter on [T ⋆

1 ] generated by C. δ(U) is an ultrafilter on

[T1] generated by Γ′′C. Γ(A0) ⊇ Γ(A1) ⊇ Γ(A2) ⊇ . . . A0 ⊇ A1 ⊇ A2 ⊇ . . . ∃A ∈ C, ∀i < ω, A \ ri(A) ⊆ Ai ∀i < ω, Γ(A) \ ri(Γ(A)) ⊆ Γ(Ai) Let F : [T1]2 → 3 be the map such that F{s, t} is the length of the longest common initial segment of δ−1(s) and δ−1(t). For each A ∈ C, F does not

mit a value on [Γ(A)]2. Therefore γ(U) is not weakly-Ramsey.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 15 / 19

SLIDE 70

Selective but not Ramsey ultrafilters Rn

Definition Let n be a positive integer. Suppose that T1, T2, . . . , Tn have been defined.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 16 / 19

SLIDE 71

Selective but not Ramsey ultrafilters Rn

Definition Let n be a positive integer. Suppose that T1, T2, . . . , Tn have been

defined. For each i < ω, let

Tn+1(i) =

i⌢ s : s ∈ Tn(j) & i(i + 1)

2 ≤ j ≤ (i + 1)(i + 2) 2

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University)

Selective for R but not Ramsey for R August 9, 2013 16 / 19

SLIDE 72

Selective but not Ramsey ultrafilters Rn

Definition Let n be a positive integer. Suppose that T1, T2, . . . , Tn have been

defined. For each i < ω, let

Tn+1(i) =

i⌢ s : s ∈ Tn(j) & i(i + 1)

2 ≤ j ≤ (i + 1)(i + 2) 2

Tn+1 =
i<ω

Tn+1(i)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 16 / 19

SLIDE 73

Selective but not Ramsey ultrafilters Rn

Definition Let n be a positive integer. Suppose that T1, T2, . . . , Tn have been

defined. For each i < ω, let

Tn+1(i) =

i⌢ s : s ∈ Tn(j) & i(i + 1)

2 ≤ j ≤ (i + 1)(i + 2) 2

Tn+1 =
i<ω

Tn+1(i) T ⋆

n+1(i) =

i⌢ s : s ∈ T ⋆

n (j) & i(i + 1)

2 ≤ j ≤ (i + 1)(i + 2) 2

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University)

Selective for R but not Ramsey for R August 9, 2013 16 / 19

SLIDE 74

Selective but not Ramsey ultrafilters Rn

Definition Let n be a positive integer. Suppose that T1, T2, . . . , Tn have been

defined. For each i < ω, let

Tn+1(i) =

i⌢ s : s ∈ Tn(j) & i(i + 1)

2 ≤ j ≤ (i + 1)(i + 2) 2

Tn+1 =
i<ω

Tn+1(i) T ⋆

n+1(i) =

i⌢ s : s ∈ T ⋆

n (j) & i(i + 1)

2 ≤ j ≤ (i + 1)(i + 2) 2

T ⋆

n+1 =

i<ω

T ⋆

n+1(i)

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 16 / 19

SLIDE 75

Selective but not Ramsey ultrafilters Rn

Theorem (T., [7]) Let n be a positive integer. If C is a (R⋆

n, ≤∗)-generic filter then Γ′′C

generates an ultrafilter on [Tn] which is selective for Rn but not Ramsey for Rn.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 17 / 19

SLIDE 76

Selective but not Ramsey ultrafilters Rn

Theorem (T., [7]) Let n be a positive integer. If C is a (R⋆

n, ≤∗)-generic filter then Γ′′C

generates an ultrafilter on [Tn] which is selective for Rn but not Ramsey for Rn.

· · ·

R2

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 17 / 19

SLIDE 77

Selective but not Ramsey ultrafilters Rn

References I

David Booth. Ultrafilters on a countable set. Annals of Mathematical Logic, 2:1–24, 1970. Natasha Dobrinen and Stevo Todorcevic. Ramsey-classification theorems and their application in the Tukey theory of ultrafilters, part 1. Transactions of the American Mathematical Society, to appear. Erik Ellentuck. A new proof that analytic sets are Ramsey. Journal of Symbolic Logic, 39:163–165, 1974. Claude Leflamme. Forcing with filters and complete combinatorics. Annals of Pure and Applied Logic, 42:125–163, 1967.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 18 / 19

SLIDE 78

Selective but not Ramsey ultrafilters Rn

References II

Jose G. Mijares. A notion of selective ultrafilter corresponding to topological Ramsey spaces.

Math. Log. Quart., 53(3):255–267, 2007.

Stevo Todorcevic. Introduction to Ramsey Spaces. Princeton University Press, 2010. Timothy Trujillo. Selective but not ramsey. Preprint, 2013.

Timothy Onofre Trujillo (University of Denver BLAST 2013 – Chapman University) Selective for R but not Ramsey for R August 9, 2013 19 / 19