Partitions properties of separable metric spaces L. Nguyen Van Th e - - PowerPoint PPT Presentation

Partitions properties of separable metric spaces

• L. Nguyen Van Th´

e

Universit´ e Aix-Marseille

May 2011

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 1 / 12

Milman’s theorem

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 2 / 12

Milman’s theorem

Sn: the unit sphere of Euclidean Rn. S∞: the unit sphere of ℓ2 (the separable, infinite-dimensional, real Hilbert space).

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 2 / 12

Milman’s theorem

Sn: the unit sphere of Euclidean Rn. S∞: the unit sphere of ℓ2 (the separable, infinite-dimensional, real Hilbert space).

Theorem (Milman, 71)

Let n > 0, ε > 0. Then there is N ∈ N such that whenever SN = R ∪ B, we have Sn ֒ → (R)ε

• r Sn ֒

→ (B)ε. In symbols: ∀n ∈ N ∃N ∈ N ∀ε > 0 SN

ε

− → (Sn)2.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 2 / 12

Milman’s theorem

Sn: the unit sphere of Euclidean Rn. S∞: the unit sphere of ℓ2 (the separable, infinite-dimensional, real Hilbert space).

Theorem (Milman, 71)

Let n > 0, ε > 0. Then there is N ∈ N such that whenever SN = R ∪ B, we have Sn ֒ → (R)ε

• r Sn ֒

→ (B)ε. In symbols: ∀n ∈ N ∃N ∈ N ∀ε > 0 SN

ε

− → (Sn)2. Remark: This is implied by:

Theorem (Matouˇ sek-R¨

• dl, 95)

Let X ⊂ S∞ finite, affinely independent, with circumradius < 1. Then there is a finite Y ⊂ S∞, affinely independent, with circumradius < 1 such that Y − → (X)2.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 2 / 12

The distortion problem

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 3 / 12

The distortion problem

Question

Does the following version of Milman’s result hold: ∀ε > 0 S∞

ε

− → (S∞)2? (Is S∞ approximately indivisible?)

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 3 / 12

The distortion problem

Question

Does the following version of Milman’s result hold: ∀ε > 0 S∞

ε

− → (S∞)2? (Is S∞ approximately indivisible?)

Theorem (Odell-Schlumprecht, 94)

No.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 3 / 12

The distortion problem

Question

Does the following version of Milman’s result hold: ∀ε > 0 S∞

ε

− → (S∞)2? (Is S∞ approximately indivisible?)

Theorem (Odell-Schlumprecht, 94)

No.

Question

Is there a direct, geometric or combinatorial, argument?

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 3 / 12

The Urysohn sphere

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 4 / 12

The Urysohn sphere

Theorem (Urysohn, 27)

Up to isometry, there is a unique complete separable ultrahomogeneous metric space into which any separable metric space embeds.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 4 / 12

The Urysohn sphere

Theorem (Urysohn, 27)

Up to isometry, there is a unique complete separable ultrahomogeneous metric space into which any separable metric space embeds.

Definition

The space above is the Urysohn space, denoted U. Up to isometry, there is a unique sphere of diameter 1 in U. The corresponding metric space is the Urysohn sphere, denoted S.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 4 / 12

The Urysohn sphere

Theorem (Urysohn, 27)

Up to isometry, there is a unique complete separable ultrahomogeneous metric space into which any separable metric space embeds.

Definition

The space above is the Urysohn space, denoted U. Up to isometry, there is a unique sphere of diameter 1 in U. The corresponding metric space is the Urysohn sphere, denoted S.

Remark

◮ For some finite approximate Ramsey type properties, U and ℓ2 behave

• similarly. So do S and S∞ (Gromov-Milman, 84 ; Pestov, 02).

◮ For exact finite Ramsey properties, the analogy is not clear yet

(Kechris-Pestov-Todorcevic, Neˇ setˇ ril, 05).

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 4 / 12

Partitions of S

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 5 / 12

Partitions of S

Theorem (Lopez-Abad - NVT - Sauer, 09)

Let ε > 0. Then: S

ε

− → (S)2.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 5 / 12

Partitions of S

Theorem (Lopez-Abad - NVT - Sauer, 09)

Let ε > 0. Then: S

ε

− → (S)2.

Corollary

Let ε > 0. Then: SC([0,1])

ε

− → (SC([0,1]))2. Note: in general, cannot require the copy to be linear.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 5 / 12

How the proof went

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 6 / 12

How the proof went

Proposition

◮ S is the unique complete separable metric space with distances in

[0, 1] into which any separable metric space with distances in [0, 1] embeds.

◮ S has a countable rational analogue: the space SQ, unique countable

ultrahomogeneous with distances in [0, 1] ∩ Q into which any countable metric space with distances in [0, 1] ∩ Q embeds.

◮ SQ embeds densely into S.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 6 / 12

How the proof went

Proposition

◮ S is the unique complete separable metric space with distances in

[0, 1] into which any separable metric space with distances in [0, 1] embeds.

◮ S has a countable rational analogue: the space SQ, unique countable

ultrahomogeneous with distances in [0, 1] ∩ Q into which any countable metric space with distances in [0, 1] ∩ Q embeds.

◮ SQ embeds densely into S.

Question

Do we have SQ − → (SQ)2? (Is SQ indivisible?)

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 6 / 12

How the proof went

Proposition

◮ S is the unique complete separable metric space with distances in

[0, 1] into which any separable metric space with distances in [0, 1] embeds.

◮ S has a countable rational analogue: the space SQ, unique countable

ultrahomogeneous with distances in [0, 1] ∩ Q into which any countable metric space with distances in [0, 1] ∩ Q embeds.

◮ SQ embeds densely into S.

Question

Do we have SQ − → (SQ)2? (Is SQ indivisible?)

Theorem (Delhomm´ e-Laflamme-Pouzet-Sauer, 07)

No.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 6 / 12

Remark

Crucial fact: the distance set of SQ is too rich.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 7 / 12

Remark

Crucial fact: the distance set of SQ is too rich.

Proposition

◮ Up to isometry, there is a unique countable ultrahomogeneous metric

space with distances in {1, . . . , m} into which every countable metric space with distances in {1, . . . , m} embeds. (Um, the Urysohn space with distances in {1, . . . , m})

◮ Um/m embeds as a 1/2m-dense subspace of S.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 7 / 12

Remark

Crucial fact: the distance set of SQ is too rich.

Proposition

◮ Up to isometry, there is a unique countable ultrahomogeneous metric

space with distances in {1, . . . , m} into which every countable metric space with distances in {1, . . . , m} embeds. (Um, the Urysohn space with distances in {1, . . . , m})

◮ Um/m embeds as a 1/2m-dense subspace of S.

Theorem (Lopez-Abad - NVT, 08)

TFAE:

• 1. ∀ε > 0 S

ε

− → (S)2.

• 2. ∀m ∈ N Um −

→ (Um)2.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 7 / 12

Remark

Crucial fact: the distance set of SQ is too rich.

Proposition

◮ Up to isometry, there is a unique countable ultrahomogeneous metric

space with distances in {1, . . . , m} into which every countable metric space with distances in {1, . . . , m} embeds. (Um, the Urysohn space with distances in {1, . . . , m})

◮ Um/m embeds as a 1/2m-dense subspace of S.

Theorem (Lopez-Abad - NVT, 08)

TFAE:

• 1. ∀ε > 0 S

ε

− → (S)2.

• 2. ∀m ∈ N Um −

→ (Um)2.

Theorem (NVT - Sauer, 09)

∀m ∈ N Um − → (Um)2.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 7 / 12

Remark

The spaces Um are particular cases of spaces of the following:

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 8 / 12

Remark

The spaces Um are particular cases of spaces of the following:

◮ For some countable sets S ⊂]0, +∞[, there is a countable

ultrahomogeneous space US with distances in S and into which every countable metric space with distances in S embeds.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 8 / 12

Remark

The spaces Um are particular cases of spaces of the following:

◮ For some countable sets S ⊂]0, +∞[, there is a countable

ultrahomogeneous space US with distances in S and into which every countable metric space with distances in S embeds.

◮ Equivalently: the class of finite metric spaces with distances in S

amalgamates.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 8 / 12

Remark

The spaces Um are particular cases of spaces of the following:

◮ For some countable sets S ⊂]0, +∞[, there is a countable

ultrahomogeneous space US with distances in S and into which every countable metric space with distances in S embeds.

◮ Equivalently: the class of finite metric spaces with distances in S

amalgamates.

◮ Those sets S have been characterized

(Delhomm´ e-Laflamme-Pouzet-Sauer, 07). Examples include:

◮ S closed under addition, or initial segment of such a set. ◮ S well founded with s+ > 2s for every s ∈ S.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 8 / 12

Remark

The spaces Um are particular cases of spaces of the following:

◮ For some countable sets S ⊂]0, +∞[, there is a countable

ultrahomogeneous space US with distances in S and into which every countable metric space with distances in S embeds.

◮ Equivalently: the class of finite metric spaces with distances in S

amalgamates.

◮ Those sets S have been characterized

(Delhomm´ e-Laflamme-Pouzet-Sauer, 07). Examples include:

◮ S closed under addition, or initial segment of such a set. ◮ S well founded with s+ > 2s for every s ∈ S.

Question

Let S ⊂]0, +∞[ be bounded so that US exists. Is US indivisible?

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 8 / 12

Indivisibility of US, S finite

◮ |S| = 1: Trivial.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 9 / 12

Indivisibility of US, S finite

◮ |S| = 1: Trivial. ◮ |S| = 2:

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 9 / 12

Indivisibility of US, S finite

◮ |S| = 1: Trivial. ◮ |S| = 2:

◮ S = {1, 2}: Random graph. Indivisible.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 9 / 12

Indivisibility of US, S finite

◮ |S| = 1: Trivial. ◮ |S| = 2:

◮ S = {1, 2}: Random graph. Indivisible. ◮ S = {1, 3}: Ultrametric. Indivisible.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 9 / 12

Indivisibility of US, S finite

◮ |S| = 1: Trivial. ◮ |S| = 2:

◮ S = {1, 2}: Random graph. Indivisible. ◮ S = {1, 3}: Ultrametric. Indivisible.

◮ |S| = 3:

◮ Essentially 6 possible distance sets S. ◮ All of them provide an indivisible space.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 9 / 12

Indivisibility of US, S finite

◮ |S| = 1: Trivial. ◮ |S| = 2:

◮ S = {1, 2}: Random graph. Indivisible. ◮ S = {1, 3}: Ultrametric. Indivisible.

◮ |S| = 3:

◮ Essentially 6 possible distance sets S. ◮ All of them provide an indivisible space.

◮ |S| = 4:

◮ Essentially 22 possible distance sets S.

Eg: {1, 2, 3, 4}, {1, 4, 6, 7}, {1, 3, 7, 10}, . . .

◮ All indivisible.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 9 / 12

Indivisibility of US, S finite

◮ |S| = 1: Trivial. ◮ |S| = 2:

◮ S = {1, 2}: Random graph. Indivisible. ◮ S = {1, 3}: Ultrametric. Indivisible.

◮ |S| = 3:

◮ Essentially 6 possible distance sets S. ◮ All of them provide an indivisible space.

◮ |S| = 4:

◮ Essentially 22 possible distance sets S.

Eg: {1, 2, 3, 4}, {1, 4, 6, 7}, {1, 3, 7, 10}, . . .

◮ All indivisible.

Theorem (Sauer, 10)

US is indivisible for every finite S.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 9 / 12

Weak indivisibility

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 10 / 12

Weak indivisibility

◮ Some spaces are not indivisible (resp. approximately indivisible). Still,

some weaker partition relations can hold.

◮ Typically, what can we say about: S∞, ℓ2, SQ, U, UQ, UN?

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 10 / 12

Weak indivisibility

◮ Some spaces are not indivisible (resp. approximately indivisible). Still,

some weaker partition relations can hold.

◮ Typically, what can we say about: S∞, ℓ2, SQ, U, UQ, UN? ◮ For finite X ⊂ S∞ and ε > 0, know that

S∞

ε

− → (X)2 Can we strengthen that to S∞

ε

− → (X, S∞)?

◮ Idem for ℓ2 and U.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 10 / 12

Weak indivisibility

◮ Some spaces are not indivisible (resp. approximately indivisible). Still,

some weaker partition relations can hold.

◮ Typically, what can we say about: S∞, ℓ2, SQ, U, UQ, UN? ◮ For finite X ⊂ S∞ and ε > 0, know that

S∞

ε

− → (X)2 Can we strengthen that to S∞

ε

− → (X, S∞)?

◮ Idem for ℓ2 and U. ◮ For finite X ⊂ SQ, know that

SQ − → (X)2 What about SQ − → (X, SQ)?

◮ Idem for UQ, UN.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 10 / 12

Answers [NVT-Sauer, 10]

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 11 / 12

Answers [NVT-Sauer, 10]

UN and U:

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 11 / 12

Answers [NVT-Sauer, 10]

UN and U:

Theorem

• 1. UN −

→ (X, UN) for every finite X ⊂ UN. In fact UN − → (Um, UN) for every m ∈ N.

• 2. U

ε

− → (X, U) for every finite X ⊂ U (even compact) and ε > 0.

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 11 / 12

Answers [NVT-Sauer, 10]

UN and U:

Theorem

• 1. UN −

→ (X, UN) for every finite X ⊂ UN. In fact UN − → (Um, UN) for every m ∈ N.

• 2. U

ε

− → (X, U) for every finite X ⊂ U (even compact) and ε > 0. UQ and SQ:

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 11 / 12

Answers [NVT-Sauer, 10]

UN and U:

Theorem

• 1. UN −

→ (X, UN) for every finite X ⊂ UN. In fact UN − → (Um, UN) for every m ∈ N.

• 2. U

ε

− → (X, U) for every finite X ⊂ U (even compact) and ε > 0. UQ and SQ:

Question (Embarrassingly open)

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 11 / 12

Answers [NVT-Sauer, 10]

UN and U:

Theorem

• 1. UN −

→ (X, UN) for every finite X ⊂ UN. In fact UN − → (Um, UN) for every m ∈ N.

• 2. U

ε

− → (X, U) for every finite X ⊂ U (even compact) and ε > 0. UQ and SQ:

Question (Embarrassingly open)

Let X be a unit distance pair of points. Does UQ − → (X, UQ)? Idem for SQ with distance δ < 1/2. The best we can do is

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 11 / 12

Answers [NVT-Sauer, 10]

UN and U:

Theorem

• 1. UN −

→ (X, UN) for every finite X ⊂ UN. In fact UN − → (Um, UN) for every m ∈ N.

• 2. U

ε

− → (X, U) for every finite X ⊂ U (even compact) and ε > 0. UQ and SQ:

Question (Embarrassingly open)

Let X be a unit distance pair of points. Does UQ − → (X, UQ)? Idem for SQ with distance δ < 1/2. The best we can do is

Theorem

UQ

ε

− → (X, UQ) for every finite X ⊂ UQ and ε > 0. (for SQ, already know from LA-NVT-S that SQ

ε

− → (SQ)2 for every ε > 0)

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 11 / 12

Two open questions

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 12 / 12

Two open questions

Question

What about weak indivisibility properties for ℓ2 and S∞?

• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 12 / 12

Two open questions

Question

What about weak indivisibility properties for ℓ2 and S∞?

Question

Is there a general finite Ramsey theorem for finite, affinely independent, metric subspaces of ℓ2? (For points: cf Frankl-R¨

• dl, 90)

Same question for S∞, with same spaces, plus requirement that circumradius < 1. (For points, cf Matouˇ sek-R¨

• dl, 95)
• L. Nguyen Van Th´

e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 12 / 12