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 S n : the unit sphere of Euclidean R n . S ∞ : the unit sphere of ℓ 2 (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 S n : the unit sphere of Euclidean R n . S ∞ : the unit sphere of ℓ 2 (the separable, infinite-dimensional, real Hilbert space). Theorem (Milman, 71) Let n > 0 , ε > 0 . Then there is N ∈ N such that whenever S N = R ∪ B, we have S n ֒ → ( R ) ε or S n ֒ → ( B ) ε . In symbols: S N → ( S n ) 2 . ε ∀ n ∈ N ∃ N ∈ N ∀ ε > 0 − L. Nguyen Van Th´ e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 2 / 12

Milman’s theorem S n : the unit sphere of Euclidean R n . S ∞ : the unit sphere of ℓ 2 (the separable, infinite-dimensional, real Hilbert space). Theorem (Milman, 71) Let n > 0 , ε > 0 . Then there is N ∈ N such that whenever S N = R ∪ B, we have S n ֒ → ( R ) ε or S n ֒ → ( B ) ε . In symbols: S N → ( S n ) 2 . ε ∀ n ∈ N ∃ N ∈ N ∀ ε > 0 − Remark: This is implied by: Theorem (Matouˇ sek-R¨ odl, 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: S ∞ → ( S ∞ ) 2 ? ε ∀ ε > 0 − (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: S ∞ → ( S ∞ ) 2 ? ε ∀ ε > 0 − (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: S ∞ → ( S ∞ ) 2 ? ε ∀ ε > 0 − (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 ℓ 2 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: ε S − → ( 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: ε S − → ( S ) 2 . Corollary Let ε > 0 . Then: ε − → ( S C ([0 , 1]) ) 2 . S C ([0 , 1]) 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 S Q , unique countable ultrahomogeneous with distances in [0 , 1] ∩ Q into which any countable metric space with distances in [0 , 1] ∩ Q embeds. ◮ S Q 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 S Q , unique countable ultrahomogeneous with distances in [0 , 1] ∩ Q into which any countable metric space with distances in [0 , 1] ∩ Q embeds. ◮ S Q embeds densely into S . Question Do we have S Q − → ( S Q ) 2 ? (Is S Q 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 S Q , unique countable ultrahomogeneous with distances in [0 , 1] ∩ Q into which any countable metric space with distances in [0 , 1] ∩ Q embeds. ◮ S Q embeds densely into S . Question Do we have S Q − → ( S Q ) 2 ? (Is S Q 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 S Q 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 S Q 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. ( U m , the Urysohn space with distances in { 1 , . . . , m } ) ◮ U m / m embeds as a 1 / 2 m-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 S Q 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. ( U m , the Urysohn space with distances in { 1 , . . . , m } ) ◮ U m / m embeds as a 1 / 2 m-dense subspace of S . Theorem (Lopez-Abad - NVT, 08) TFAE: ε 1. ∀ ε > 0 S − → ( S ) 2 . 2. ∀ m ∈ N U m − → ( U m ) 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 S Q 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. ( U m , the Urysohn space with distances in { 1 , . . . , m } ) ◮ U m / m embeds as a 1 / 2 m-dense subspace of S . Theorem (Lopez-Abad - NVT, 08) TFAE: ε 1. ∀ ε > 0 S − → ( S ) 2 . 2. ∀ m ∈ N U m − → ( U m ) 2 . Theorem (NVT - Sauer, 09) ∀ m ∈ N U m − → ( U m ) 2 . L. Nguyen Van Th´ e (Universit´ e Aix-Marseille) Partitions of separable metric spaces May 2011 7 / 12

Remark The spaces U m 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