Jordi Lopez-Abad

32 Pages·2015·0.24 MB·English

Approximate Ramsey properties of finite dimensional
normed spaces.
J. Lopez-Abad
InstitutodeCienciasMatem´aticas,CSIC,Madrid
IME,USP.
ResearchsupportedbytheFAPESPproject13/24827-1
February 23th, 2015
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Outline
1 (Approximate) Ramsey Properties
Structural Ramsey Theorems
Classical sequence spaces (cid:96)n’s
p
Borsuk-Ulam Theorem
2 Applications. Extreme Amenability
Extreme Amenability
L´evy groups, Concentration of measure
Applications
3 Partitions; Dual Ramsey and concentration of measure
The case p = ∞; Dual Ramsey Theorem
An open problem
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(Approximate)RamseyProperties StructuralRamseyTheorems
Notation: [n] := {1,··· ,n}. Recall the well-know Ramsey Theorem:
Given integers d,m and r there is an integer n such that for every coloring
c : [n]d := {s ⊆ [n] : #s = d} → [r] (1)
there is
s ∈ [n]m (2)
such that
c (cid:22) [s]d is constant. (3)
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(Approximate)RamseyProperties StructuralRamseyTheorems
This can be rephrased as follows: Let A = (A,< ) and B = (B,< ) be
A B
two finite linearly ordered sets and let r ∈ N. Then there exists
C = (C,< ) such that for every coloring
C
(cid:18) (cid:19)
C
c : := {A(cid:48) ⊆ C : (A(cid:48),< ) and (A,< ) are order-isomorphic} → [r]
C A
A
(4)
there is
(cid:18) (cid:19)
C
B(cid:48) ∈ (5)
B
such that
(cid:18)B(cid:48)(cid:19)
c (cid:22) is constant. (6)
A
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(Approximate)RamseyProperties StructuralRamseyTheorems
So, given a family K of structures of the same sort, we say that K has the
Ramsey Property when for every A,B ∈ K and r ∈ N there exists C ∈ K
such that for every coloring
(cid:18) (cid:19)
C
c : := {A(cid:48) ⊆ C : A(cid:48) ∼= A} → [r] (7)
A
there is
(cid:18) (cid:19)
C
B(cid:48) ∈ (8)
B
such that
(cid:18)B(cid:48)(cid:19)
c (cid:22) is constant. (9)
A
We will abbreviate this by
C → (B)A (10)
r
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(Approximate)RamseyProperties StructuralRamseyTheorems
Examples
Example
The class of all finite ordered Graphs has the Ramsey property (Nesetril
and Rodl).
Example
The class of finite-dimensional vector spaces over a finite field has the
Ramsey property (Graham, Leeb and Rothschild)
Example
The class of all finite ordered metric spaces has the Ramsey property
(Nesetril).
Example
The class of naturally ordered finite boolean algebras is Ramsey (Graham
and Rothschild, Dual Ramsey Theorem)
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(Approximate)RamseyProperties Classicalsequencespaces(cid:96)np’s
Definition
Let 1 ≤ p ≤ ∞, n ∈ N. The p-norm (cid:107)·(cid:107) on Rn is defined for (a ) by
p i i<n
(cid:107)(ai)i<n(cid:107)p :=((cid:88)|ai|p)p1 for p < ∞ (11)
i<n
(cid:107)(a ) (cid:107) :=max|a |. (12)
i i<n ∞ i
i<n
Definition
Given two Banach spaces X and Y, by a (linear isometric) embedding
from X into Y we mean a linear operator T : X → Y such that
(cid:107)T(x)(cid:107) = (cid:107)x(cid:107) for all x ∈ X. (13)
Y X
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(Approximate)RamseyProperties Classicalsequencespaces(cid:96)np’s
Definition
Let
Emb(X,Y) (14)
be the collection of all embeddings from X into Y, and let
(cid:18) (cid:19)
Y
:= {Z ⊆ Y : Z is isometric to X}. (15)
X
Note that Emb(X,Y) is a metric space with the norm distance
d(T,U) := (cid:107)T −U(cid:107) := sup (cid:107)T(x)−U(x)(cid:107). (16)
x∈S
X
When Y is finite dimensional (cid:0)Y(cid:1) is also a metric space when considering
X
the Hausdorff distance between the unit balls of copies X(cid:48) and X(cid:48)(cid:48) of X in
Y.
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(Approximate)RamseyProperties Classicalsequencespaces(cid:96)np’s
The approximate Ramsey property would be: For every 1 ≤ p ≤ ∞ every
integers d,m and r there is n such that for every coloring
(cid:18)(cid:96)n(cid:19)
c : p → [r] (17)
(cid:96)d
p
there exist
(cid:18)(cid:96)n(cid:19)
X ∈ p and 1 ≤ i ≤ r (18)
(cid:96)m
p
such that
(cid:18) (cid:19)
X
⊆ (c−1(i)) . (19)
(cid:96)d ε
p
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(Approximate)RamseyProperties Classicalsequencespaces(cid:96)np’s
In fact we have a more demanding notion.
Definition
Given 1 ≤ p ≤ ∞, integers d,m and r and ε > 0, let n (d,m,r,ε) be the
p
minimal integer n (if exists) such that for every coloring
c : Emb((cid:96)d,(cid:96)n) → [r] (20)
p p
there exist
γ ∈ Emb((cid:96)m,(cid:96)n) and 1 ≤ i ≤ r (21)
p p
such that
γ ◦Emb((cid:96)d,(cid:96)m) ⊆ (c−1(i)) . (22)
p p ε
(cid:0)(cid:96)n(cid:1)
This property implies the first one about p .
(cid:96)d
p
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