Sequences in Large Spaces

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Sequences in Large Spaces

Stevo Todorcevic

C.N.R.S. Paris and University of Toronto

Maresias, 28 August 2014

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Outline

1. Weakly Null Sequences

2. Ramsey Theory of Fronts and Barriers 3. w^∗-null sequences and the quotient problem 4. P-ideal Dichotomy

5. Weakly-null sequence on Polish spaces 6. Ramsey Theory on Trees

7. Unconditional sequences

8. The Free-Set Property and the Product Ramsey property 9. Long Conditional Weakly-null Sequences

10. Positional Graphs 11. Subsymmetric sequences 12. Compact families on trees

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Outline

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Outline

2. Ramsey Theory of Fronts and Barriers

3. w^∗-null sequences and the quotient problem 4. P-ideal Dichotomy

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Outline

2. Ramsey Theory of Fronts and Barriers 3. w^∗-null sequences and the quotient problem

4. P-ideal Dichotomy

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Outline

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Outline

5. Weakly-null sequence on Polish spaces

6. Ramsey Theory on Trees 7. Unconditional sequences

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Outline

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Outline

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Outline

8. The Free-Set Property and the Product Ramsey property

9. Long Conditional Weakly-null Sequences 10. Positional Graphs

11. Subsymmetric sequences 12. Compact families on trees

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Outline

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Outline

10. Positional Graphs

11. Subsymmetric sequences 12. Compact families on trees

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Outline

10. Positional Graphs 11. Subsymmetric sequences

12. Compact families on trees

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Outline

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Structure in weakly null sequences

Theorem (Bessage-Pelczynski, 1958)

For everyε >0,every normalized weakly null sequence(x_n) contains an infinite(1 +ε)-basic subsequence(xni).

Theorem (Maurey-Rosenthal, 1977)

(1) For every ε >0 and every α < ω^ω,every normalized weakly null sequence inC(α+ 1)has a(2 +ε)-unconditional subsequence.

(2) For every ε >0 every normalized weakly null sequence in C(ω^ω+ 1)has a(4 +ε)-unconditional subsequence.

(3) There is a normalized weakly null sequence in C(ω^ω²+ 1)with no unconditional subsequence.

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Structure in weakly null sequences

For everyε >0,every normalized weakly null sequence (x_n) contains an infinite(1 +ε)-basic subsequence(xni).

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Structure in weakly null sequences

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Structure in weakly null sequences

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Structure in weakly null sequences

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Finite and partial unconditionality

Theorem (Odell 1993, Dodos-LopezAbad-Todorcevic, 2011) Let k a positive integer andε >0. Suppose that for every i <k we are given a normalized weakly null sequence(x_nⁱ)^∞_n=0 in some Banach space X . Then, there exists an infinite set M of integers such that for every{n₀ <· · ·<nk−1} ⊆M the k-sequence (x_nⁱ_i)_i<k is (1 +ε)-unconditional.

Theorem (Arvanitakis 2006, Gasparis-Odell-Wahl, 2006 Todorcevic 2005)

Suppose that(x_n)is a normalized weakly-null sequence in `∞(Γ) with the property that

inf{|x_n(γ)|:n∈N, γ ∈Γ}>0.

Then(xn) contains an infinite unconditional basic subsequence.

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Finite and partial unconditionality

Theorem (Odell 1993, Dodos-LopezAbad-Todorcevic, 2011) Let k a positive integer andε >0. Suppose that for every i <k we are given a normalized weakly null sequence(x_nⁱ)^∞_n=0 in some Banach space X . Then, there exists an infinite set M of integers such that for every{n₀ <· · ·<nk−1} ⊆M the k-sequence (x_nⁱ_i)_i<k is (1 +ε)-unconditional.

Theorem (Arvanitakis 2006, Gasparis-Odell-Wahl, 2006 Todorcevic 2005)

Suppose that(x_n)is a normalized weakly-null sequence in `∞(Γ) with the property that

inf{|x_n(γ)|:n∈N, γ ∈Γ}>0.

Then(xn) contains an infinite unconditional basic subsequence.

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Theorem (Elton 1978)

For every0< ε≤1 there is a constant C(ε)≥1 such that every normalized weakly null sequence(xn) has an infinite subsequence (xn_i)such that

kX

i∈I

aixnik ≤C(ε)kX

j∈J

ajxnjk

for every pair I ⊆J of subsets ofNand every choice (aj :j ∈J) of scalars such thatε≤ |a_j| ≤1for all j ∈J.

Problem (Elton unconditionality constant problem) Issup_0<ε≤1C(ε)<∞?

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Theorem (Elton 1978)

For every0< ε≤1 there is a constant C(ε)≥1 such that every normalized weakly null sequence(xn) has an infinite subsequence (xn_i)such that

kX

i∈I

aixnik ≤C(ε)kX

j∈J

ajxnjk

for every pair I ⊆J of subsets ofNand every choice (aj :j ∈J) of scalars such thatε≤ |a_j| ≤1for all j ∈J.

Problem (Elton unconditionality constant problem) Issup_0<ε≤1C(ε)<∞?

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Nash-Willimas’s theory of fronts and barriers

Definition (Nash-Williams 1965)

For a familyF of finite subsets ofN we say that:

1. F is thinwhenever s 6vt for s 6=t in F.

2. F is a front ifF is thin and if every infinite subset of N contains an initial segment in F.

3. F is a barrier ifF is a front and ifs 6⊆t for s 6=t inF.

Theorem (Nash-Williams 1965)

SupposeH=H₀∪ · · · ∪ H_l is a finite partition of a thin family H of finite subsets ofN.Then there is an infinite set M⊆N and i <l such thatHM ⊆ H_i,where

HM ={s ∈ H:s ⊆M}.

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Nash-Willimas’s theory of fronts and barriers

HM ={s ∈ H:s ⊆M}.

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Nash-Willimas’s theory of fronts and barriers

HM ={s ∈ H:s ⊆M}.

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Nash-Willimas’s theory of fronts and barriers

HM ={s ∈ H:s ⊆M}.

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Nash-Willimas’s theory of fronts and barriers

SupposeH=H₀∪ · · · ∪ H_l is a finite partition of a thin familyH of finite subsets ofN.Then there is an infinite set M⊆N and i <l such thatHM ⊆ H_i,where

HM ={s ∈ H:s ⊆M}.

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Theorem (Pudlak-R¨odl 1982)

For every frontB onNand every mapping f :B →Nthere exist an infinite subset M ofNand a mapping ϕ:BM →[N]^<∞ such that:

(1) ϕis an internal mapping, i.e.,ϕ(s)⊆s for all ∈ BM, (2) the range of ϕis a thin family, i.e.,ϕ(s)6vϕ(t) for all

s,t ∈ BM such that ϕ(s)6=ϕ(t),and (3) for s,t ∈ BM,f(s) =f(t) iff ϕ(s) =ϕ(t).

Remark

There can be only one mappingϕ:BM →[N]^<∞ satisfying the conditions (1), (2) and (3) from the Theorem on a given infinite subsetM ofN.

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(1) ϕis an internal mapping, i.e.,ϕ(s)⊆s for all ∈ BM,

(2) the range of ϕis a thin family, i.e.,ϕ(s)6vϕ(t) for all s,t ∈ BM such that ϕ(s)6=ϕ(t),and

(3) for s,t ∈ BM,f(s) =f(t) iff ϕ(s) =ϕ(t).

Remark

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s,t ∈ BM such that ϕ(s)6=ϕ(t),and

(3) for s,t ∈ BM,f(s) =f(t) iff ϕ(s) =ϕ(t).

Remark

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s,t ∈ BM such that ϕ(s)6=ϕ(t),and (3) for s,t∈ B M,f(s) =f(t) iff ϕ(s) =ϕ(t).

Remark

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s,t ∈ BM such that ϕ(s)6=ϕ(t),and (3) for s,t∈ B M,f(s) =f(t) iff ϕ(s) =ϕ(t).

Remark

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w

^∗

-null sequences and the quotient problem

Theorem (Josefson 1975, Nissenzweig 1975)

For every infinite-dimensional normed space X there is a normalized w^∗-null sequence (fn)^∞_n=0 in X^∗.

Theorem (Johnson-Rosenthal 1972)

Every separable infinite-dimensional space has an infinite-dimensional quotient with a Schauder basis. Theorem (Todorcevic 2006)

Suppose that a Banach space X has density<mand that its dual X^∗ has an uncountable normalized w^∗-null sequence. Then X has a quotient with a Schauder basis of lengthω₁.

Remark

Recall thatmis the Baire-category number of the class of compact Hausdorff spaces satisfying thecountable chain condition.

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w

^∗

-null sequences and the quotient problem

Every separable infinite-dimensional space has an infinite-dimensional quotient with a Schauder basis. Theorem (Todorcevic 2006)

Remark

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w

^∗

-null sequences and the quotient problem

Every separable infinite-dimensional space has an infinite-dimensional quotient with a Schauder basis.

Theorem (Todorcevic 2006)

Remark

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w

^∗

-null sequences and the quotient problem

Remark

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w

^∗

-null sequences and the quotient problem

Remark

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P-ideal dichotomy, PID

Definition

Anidealon an index set S is simply a familyI of subsets of S closed under taking subsets and finite unions of its elements. We shall consider only ideals ofcountable subsets ofS and assume that all our ideals include the ideal of all finite subsets ofS. We say that such an idealI is a P-idealif for every sequence (x_n) inI there is y∈ I such that xn\y is finite for all n.

Example

1. The ideal [S]^≤ℵ⁰ of all countable subsets ofS is a P-ideal. 2. Given a family F of cardinality <b the ideal

F^⊥={x ∈[S]^≤ℵ⁰ : (∀Y ∈ F)|x∩Y|<ℵ₀} is a P-ideal.

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P-ideal dichotomy, PID

Definition

Anidealon an index set S is simply a familyI of subsets of S closed under taking subsets and finite unions of its elements.

We shall consider only ideals ofcountable subsets ofS and assume that all our ideals include the ideal of all finite subsets ofS. We say that such an idealI is a P-idealif for every sequence (x_n) inI there is y∈ I such that xn\y is finite for all n.

Example

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P-ideal dichotomy, PID

Definition

We shall consider only ideals ofcountable subsets ofS and assume that all our ideals include the ideal of all finite subsets ofS.

We say that such an idealI is a P-idealif for every sequence (x_n) inI there is y∈ I such that xn\y is finite for all n.

Example

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P-ideal dichotomy, PID

Definition

We say that such an idealI is a P-idealif for every sequence (x_n) inI there is y∈ I such that xn\y is finite for alln.

Example

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P-ideal dichotomy, PID

Definition

Example

1. The ideal [S]^≤ℵ⁰ of all countable subsets ofS is a P-ideal.

2. Given a family F of cardinality <b the ideal

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P-ideal dichotomy, PID

Definition

Example

1. The ideal [S]^≤ℵ⁰ of all countable subsets ofS is a P-ideal.

2. Given a family F of cardinality<b the ideal

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Definition

TheP-ideal dichotomyis the statement that for every P-ideal I on some index setS either

(1) there is uncountable T ⊆S such that [T]^ℵ⁰ ⊆ I,or (2) there is a countable decomposition S =S

n<ωS_n such that Sn⊥ I for all n.

Remark

1. It is known that PID follows from the strong Baire category principles such asmm> ω1.

2. It is also known that PID is consistent with GCH. 3. It is known that PID implies, for example, the Souslin

Hypothesis.

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Definition

(1) there is uncountable T ⊆S such that [T]^ℵ⁰ ⊆ I,or

(2) there is a countable decomposition S =S

Remark

Hypothesis.

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Definition

Remark

Hypothesis.

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Definition

Remark

Hypothesis.

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Definition

Remark

2. It is also known that PID is consistent with GCH.

3. It is known that PID implies, for example, the Souslin Hypothesis.

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Definition

Remark

2. It is also known that PID is consistent with GCH.

3. It is known that PID implies, for example, the Souslin Hypothesis.

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Assume PID. Let X be a nonseparable Banach space of density

<p.Then X^∗ has an uncountable normalized w^∗-null sequence.

Remark

Recall thatp is the minimal cardinality of a familyF of infinite subsets ofNsuch that the intersection of every subfamily ofF is infinite but there is no infinite setM ⊆Nsuch that M\N is finite for allN ∈ F.

Corollary (Todorcevic 2006)

Assume PID andm> ω₁.Then every non-separable Banach space has un uncountable biorthogonal system.

Assume PID andm> ω1.Then every non-separable Banach space has closed convex subset supported by all of its points.

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Remark

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Remark

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Remark

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Remark

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PID and Asplund spaces

Definition

AnAsplund space, or astrong differentiability space is a Banach spaceX with the property that every continuous convex functionf :U →Ron an open convex domainU ⊆X is Fr´echet differentiable in every point of a denseG_δ-subset ofU.

Remark

This is a well studied class of spaces with many pleasant properties such as theprojectional resolution of the identity of its dual space, thenorm-fragmentability of the w^∗-topologyof the dual ball,separability of the dual of every separable subspace, etc.

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PID and Asplund spaces

Definition

AnAsplund space, or astrong differentiability space is a Banach spaceX with the property that every continuous convex functionf :U →Ron an open convex domainU ⊆X is Fr´echet differentiable in every point of a denseG_δ-subset ofU.

Remark

This is a well studied class of spaces with many pleasant properties such as theprojectional resolution of the identityof its dual space, thenorm-fragmentability of thew^∗-topologyof the dual ball,separability of the dual of every separable subspace, etc.

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Ifb=ω1 then there is an Asplund space with no uncountable ε-biorthogonal system for any 0≤ε <1.

Recall thatb is the minimal cardinality of a subset ofN^N that is unbounded in the ordering of eventual dominance.

Theorem (Brech-Todorcevic 2012)

Assume PID. Let X be nonseparable Asplund space of density<b. Then X^∗ has an uncountable normalized w^∗-null sequence.

Corollary

Assume PID. The following are equivalent:

1. Every non-separable Asplund space has an uncountable ε-biorthogonal system for every ε >0.

2. b=ℵ₂.

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Assume PID. Let X be nonseparable Asplund space of density<b. Then X^∗ has an uncountable normalized w^∗-null sequence.

Corollary

2. b=ℵ₂.

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Assume PID. Let X be nonseparable Asplund space of density<b.

Then X^∗ has an uncountable normalized w^∗-null sequence.

Corollary

2. b=ℵ₂.

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Corollary

2. b=ℵ₂.

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Corollary

2. b=ℵ₂.

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Weakly null sequences on Polish spaces

Theorem (Mycielski 1964)

Suppose Mn⊆X^kⁿ(n= 0,1, . . .) is a sequence of subsets of the finite powers of some fixed Polish space X and suppose that M_n is a meager subset of X^kⁿ for all n. Then there is a perfect set P ⊆X such that[P]^kⁿ∩Mn=∅ for all n.

Theorem (Argyros-Dodos-Kanellopoulos 2008)

Suppose X is a Polish space and that(f_a)_a∈2N is a bounded sequence in`∞(X) such that (x,a)7→fa(x) is a Borel function from X ×2^N into Rand that

|{a∈2^N:fa(x)6= 0}| ≤ ℵ₀ for all x ∈X.

Then there is a perfect set P⊆2^Nsuch that the sequence (f_a)a∈P

is 1-unconditional.

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Weakly null sequences on Polish spaces

Suppose M_n⊆X^kⁿ(n = 0,1, . . .) is a sequence of subsets of the finite powers of some fixed Polish space X and suppose that M_n is a meager subset of X^kⁿ for all n. Then there is a perfect set P ⊆X such that[P]^kⁿ∩M_n=∅ for all n.

Suppose X is a Polish space and that(f_a)_a∈2N is a bounded sequence in`∞(X) such that (x,a)7→fa(x) is a Borel function from X ×2^N into Rand that

|{a∈2^N:fa(x)6= 0}| ≤ ℵ₀ for all x ∈X.

is 1-unconditional.

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Weakly null sequences on Polish spaces

Suppose M_n⊆X^kⁿ(n = 0,1, . . .) is a sequence of subsets of the finite powers of some fixed Polish space X and suppose that M_n is a meager subset of X^kⁿ for all n. Then there is a perfect set P ⊆X such that[P]^kⁿ∩M_n=∅ for all n.

Suppose X is a Polish space and that(f_a)_a∈2N is a bounded sequence in`∞(X) such that (x,a)7→fa(x) is a Borel function from X×2^N into Rand that

|{a∈2^N:fa(x)6= 0}| ≤ ℵ₀ for all x ∈X.

is 1-unconditional.

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Ramsey Theory of trees

Definition

Fix a rooted finitely branching treeU with no terminal nodes. A subtreeT of U will be called astrong subtree if the levels of T are subsets of the levels ofU and if for everyt∈T every

immediate successor oft inU is extended by a unique immediate successor oft inT.

Theorem (Halpern-L¨auchli 1966)

For every sequence U₀, ...,Ud−1 of rooted finitelly branching trees with no terminal nodes and for every finite colouring of thelevel productU0⊗ · · · ⊗Ud−1,we can find for each i <d a strong subtree T_i of U_i such that the T_i’s share the same level set and such that the level product T0⊗ · · · ⊗Td−1 is monochromatic. Theorem (Miliken 1981)

For every finite Borel colouring of the spaceS_∞(U) of all strong subtrees of U there is a strong subtree T of U such that the set S_∞(T)of strong subtrees of T is monochromatic.

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Ramsey Theory of trees

Definition

For every sequence U₀, ...,Ud−1 of rooted finitelly branching trees with no terminal nodes and for every finite colouring of thelevel productU0⊗ · · · ⊗Ud−1,we can find for each i <d a strong subtree T_i of U_i such that the T_i’s share the same level set and such that the level product T0⊗ · · · ⊗Td−1 is monochromatic. Theorem (Miliken 1981)

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Ramsey Theory of trees

Definition

For every sequence U₀, ...,Ud−1 of rooted finitelly branching trees with no terminal nodes and for every finite colouring of thelevel productU₀⊗ · · · ⊗Ud−1,we can find for each i <d a strong subtree T_i of U_i such that the T_i’s share the same level set and such that the level product T0⊗ · · · ⊗Td−1 is monochromatic.

Theorem (Miliken 1981)

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Ramsey Theory of trees

Definition

For every sequence U₀, ...,Ud−1 of rooted finitelly branching trees with no terminal nodes and for every finite colouring of thelevel productU₀⊗ · · · ⊗Ud−1,we can find for each i <d a strong subtree T_i of U_i such that the T_i’s share the same level set and such that the level product T0⊗ · · · ⊗Td−1 is monochromatic.

Theorem (Miliken 1981)

For every finite Borel colouring of the spaceS_∞(U)of all strong subtrees of U there is a strong subtree T of U such that the set S_∞(T)of strong subtrees of T is monochromatic.

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A parametrized Ramsey theorem for perfect sets

Theorem

For every countable Borel colouring of the product[2^N]²×[N]^∞ with colours that are invariant under finite changes on the second coordinate, there is a perfect set P ⊆2^N and an infinite set M ⊆N such that the product[P]²×[M]^∞ is monochromatic.

Example (Pol 1986)

Pol’s compact set of Baire class-1 function is represented as P= 2^<^N∪2^N∪ {∞},

where the points of the Cantor tree 2^<^N are isolated, the nodes of a branch of this tree converge to the corresponding member of 2^N and∞ is the point that compactifies the rest of the space.

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A parametrized Ramsey theorem for perfect sets

Theorem

Example (Pol 1986)

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A parametrized Ramsey theorem for perfect sets

Theorem

Example (Pol 1986)

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Suppose K is a separable compact set of Baire class-1 functions defined on some Polish space X . Let D be a countable dense subset of K,and let f be a point of K that is not G_δ in K.Then there is a homeomorphic embedding

Φ :P→K such thatΦ(∞) =f andΦ[2^<^N]⊆D.

Theorem (Argyros-Dodos-Kanellopoulos 2008) Every infinite-dimensional dual Banach space has an infinite-dimensional quotient with a Schauder basis.

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Suppose K is a separable compact set of Baire class-1 functions defined on some Polish space X . Let D be a countable dense subset of K,and let f be a point of K that is not G_δ in K.Then there is a homeomorphic embedding

Φ :P→K such thatΦ(∞) =f andΦ[2^<^N]⊆D.

Theorem (Argyros-Dodos-Kanellopoulos 2008) Every infinite-dimensional dual Banach space has an infinite-dimensional quotient with a Schauder basis.

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Unconditional sequences

Theorem (Gowers-Maurey, 1993)

There is a separable reflexive infinite-dimensional space X with no infinite unconditional basic sequence.

Theorem (Argyros-LopezAbad-Todorcevic, 2006)

There is also a non-separable reflexive space X with no infinite unconditional basic sequence.

Problem

Is there a reflexive space of density>ℵ₁ without an infinite unconditional basic sequence?

Theorem (Argyros-Tolias, 2004)

There is a space X of density2^ℵ⁰ with no infinite unconditional basic sequence.

Problem

Is there a space of density>2^ℵ⁰ without an infinite unconditional basic sequence?

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Unconditional sequences

Problem

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Unconditional sequences

Problem

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Unconditional sequences

Problem

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Unconditional sequences

Problem

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Unconditional sequences

Problem

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The unconditional basic sequence problem, revisited

Problem

1. When does an infinite-dimensional normed space contain an infinite unconditional basic sequence?

2. When does an infinite normalized weakly null sequence in some normed space contains an infinite unconditional subsequence?

Theorem (Johnson-Rosenthal 1972, Hagler-Johnson 1977) If the dual X^∗ of some Banach space X contains an infinite unconditional basic sequence then X admits a quotient with an unconditional basis.

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The unconditional basic sequence problem, revisited

Problem

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The unconditional basic sequence problem, revisited

Problem

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The unconditional basic sequence problem, revisited

Problem

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Free-set Property and Product-Ramsey Property

Definition

We say that an index-set Γ has thefree-set property, and write FSP(Γ), if every algebra A on Γ with no more than countably many operations has an infinitefree set, an infinite subsetX of Γ such that nox ∈X is in the sub algebra ofAgenerated by X\ {x}.

Definition

We say that Γ has theproduct-Ramsey property, and write PRP(Γ), if for every colouring

χ: Γ^<ω →2

of the set of all finite sequences of the index-set Γ into 2 colours there exists an infinite sequence (Xi) of 2-element subsets of Γ or, equivalently, an infinite sequence (X_i) ofinfinite subsets of Γ, such thatχ is constant onQ

i<nX_i for all n.

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Free-set Property and Product-Ramsey Property

Definition

χ: Γ^<ω →2

i<nX_i for all n.

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Free-set Property and Product-Ramsey Property

Definition

χ: Γ^<ω →2

i<nXi for all n.

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Proposition

PRP(Γ)impliesFSP(Γ)but not vice versa.

Theorem (Erd˝os-Hajnal 1966)

The free-set property fails for index-sets of cardinalities<ℵ_ω. Theorem (Koepke 1984, DiPrisco-Todorcevic 1999) The following are equiconsistent:

1. FSP(ℵ_ω). 2. PRP(ℵ_ω).

3. There is an index set supporting a non-principal countably complete ultrafilter.

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Proposition

The free-set property fails for index-sets of cardinalities<ℵ_ω.

Theorem (Koepke 1984, DiPrisco-Todorcevic 1999) The following are equiconsistent:

1. FSP(ℵ_ω). 2. PRP(ℵ_ω).

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Proposition

The free-set property fails for index-sets of cardinalities<ℵ_ω. Theorem (Koepke 1984, DiPrisco-Todorcevic 1999) The following are equiconsistent:

1. FSP(ℵ_ω).

2. PRP(ℵ_ω).

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Two-dimensional Product Ramsey Property

Definition

We say that an index-set Γ has the 2-dimensional polarized Ramsey property) and writePRP₂(Γ), if for every colouring

χ: ([Γ]²)^<ω→2

of all finite sequences of 2-element subsets of the index-set Γ into 2 colours there exist an infinite sequence (X_i) of infinitesubsets of Γ such thatχ is constant on Q

i<n[X_i]² for all n. Theorem (Shelah 1980)

PRP2(ℵ_ω) is consistent relative to the existence of infinitely many compact cardinals.

Question

What is the equiconsistency result here?

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Two-dimensional Product Ramsey Property

Definition

χ: ([Γ]²)^<ω→2

i<n[X_i]² for all n.

Theorem (Shelah 1980)

Question

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Two-dimensional Product Ramsey Property

Definition

χ: ([Γ]²)^<ω→2

i<n[X_i]² for all n.

Theorem (Shelah 1980)

Question