Approximate Ramsey properties of finite dimensional normed spaces. J. Lopez-Abad Instituto de Ciencias Matem´ aticas,CSIC, Madrid IME, USP. Research supported by the FAPESP project 13/24827-1 February 23th, 2015 J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 1 / 32
Outline 1 (Approximate) Ramsey Properties Structural Ramsey Theorems Classical sequence spaces ℓ n p ’s 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 J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 2 / 32
(Approximate) Ramsey Properties Structural Ramsey Theorems 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 ↾ [ s ] d is constant . (3) J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 3 / 32
(Approximate) Ramsey Properties Structural Ramsey Theorems This can be rephrased as follows: Let A = ( A , < A ) and B = ( B , < B ) be two finite linearly ordered sets and let r ∈ N . Then there exists C = ( C , < C ) such that for every coloring � C � := { A ′ ⊆ C : ( A ′ , < C ) and ( A , < A ) are order-isomorphic } → [ r ] c : A (4) there is � C � B ′ ∈ (5) B such that � B ′ � c ↾ is constant . (6) A J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 4 / 32
(Approximate) Ramsey Properties Structural Ramsey Theorems 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 � C � := { A ′ ⊆ C : A ′ ∼ c : = A } → [ r ] (7) A there is � C � B ′ ∈ (8) B such that � B ′ � c ↾ is constant . (9) A We will abbreviate this by C → ( B ) A (10) r J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 5 / 32
(Approximate) Ramsey Properties Structural Ramsey Theorems 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 ) J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 6 / 32
Classical sequence spaces ℓ n (Approximate) Ramsey Properties p ’s Definition Let 1 ≤ p ≤ ∞ , n ∈ N . The p-norm � · � p on R n is defined for ( a i ) i < n by 1 � | a i | p ) p for p < ∞ � ( a i ) i < n � p :=( (11) i < n � ( a i ) i < n � ∞ := max i < n | a i | . (12) 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 � T ( x ) � Y = � x � X for all x ∈ X . (13) J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 7 / 32
Classical sequence spaces ℓ n (Approximate) Ramsey Properties p ’s Definition Let Emb ( X , Y ) (14) be the collection of all embeddings from X into Y , and let � 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 ) := � T − U � := sup � T ( x ) − U ( x ) � . (16) x ∈ S X � Y � When Y is finite dimensional is also a metric space when considering X the Hausdorff distance between the unit balls of copies X ′ and X ′′ of X in Y . J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 8 / 32
Classical sequence spaces ℓ n (Approximate) Ramsey Properties p ’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 � ℓ n � p c : → [ r ] (17) ℓ d p there exist � ℓ n � p X ∈ and 1 ≤ i ≤ r (18) ℓ m p such that � X � ⊆ ( c − 1 ( i )) ε . (19) ℓ d p J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 9 / 32
Classical sequence spaces ℓ n (Approximate) Ramsey Properties p ’s In fact we have a more demanding notion. Definition Given 1 ≤ p ≤ ∞ , integers d , m and r and ε > 0 , let n p ( d , m , r , ε ) be the minimal integer n (if exists) such that for every coloring c : Emb ( ℓ d p , ℓ n p ) → [ r ] (20) there exist γ ∈ Emb ( ℓ m p , ℓ n p ) and 1 ≤ i ≤ r (21) such that p ) ⊆ ( c − 1 ( i )) ε . γ ◦ Emb ( ℓ d p , ℓ m (22) � ℓ n � This property implies the first one about p . ℓ d p J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 10 / 32
Classical sequence spaces ℓ n (Approximate) Ramsey Properties p ’s u n = ( u i ) i < n be the unit basis of R n . Given Another reformulation: Let ¯ 1 ≤ p ≤ ∞ and m ≤ n let m , n := { A ∈ M n , m : A is the matrix in the unit bases of R d and R n I p of an isometric embedding } . Then n p ( d , m , r , ε ) is the minimal integer n (if exists) such that for every coloring c : I d d , m → [ r ] (23) there exist A ∈ I p m , n and 1 ≤ i ≤ r (24) such that A · I p d , m ⊆ ( c − 1 ( i )) ε . (25) J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 11 / 32
Classical sequence spaces ℓ n (Approximate) Ramsey Properties p ’s Proposition A ∈ I ∞ m , n if and only if for every column vector c of A one has that � c � ∞ = 1 and for every row vector r of A one has that � r � 1 ≤ 1 . Proposition A ∈ I 2 m , n if and only if the sequence ( c i ) i < m of columns of A is orthonormal. Proposition Given 1 ≤ p < ∞ , p � = 2 , A ∈ I p m , n if and only if for every column vector c of A one has that � c � p = 1 and every two column vectors have disjoint support. Theorem n p ( d , m , r , ε ) exists. J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 12 / 32
(Approximate) Ramsey Properties Borsuk-Ulam Theorem The intention is to relate our result with the Borsuk-Ulam Theorem. Recall that one of the several equivalent versions ( Lusternik-Schnirelmann Theorem) of the Borsuk-Ulam theorem states that if the unit sphere S n of ℓ n +1 is covered by n + 1 many open sets, then one of them contains a 2 point x and its antipodal − x . Definition Let ( X , d ) be a metric space, ε > 0 . We say that an open covering U of X is ε -fat when for every U ∈ U there is V U open such that ( V U ) ε ⊆ U and { V U } U ∈U is still a covering of X. It is not difficult to see that if X is compact, then every open covering is ε -fat for some ε > 0. J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 13 / 32
(Approximate) Ramsey Properties Borsuk-Ulam Theorem Now the previous Theorem on embeddings can be restated as follows Theorem For every 1 ≤ p ≤ ∞ , every integers d , m and r and every ε there is some n p ( d , m , r , ε ) such that for every ε -fat open covering U of I p d , n with cardinality at most r there exists some A ∈ I p m , n such that A · I p d , m ⊆ U for some U ∈ U . (26) For example, Borsuk-Ulam Theorem is the statement n 2 (1 , 1 , r , ε ) = r for all ε > 0 , (27) because I 2 1 , n consists on 1-column-matrices ( v ) of vectors v of the sphere of ℓ n 2 , and I 2 1 , 1 = { (1) , ( − 1) } , so ( v ) · I 2 1 , 1 = { ( − v ) , ( − v ) } . J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 14 / 32
(Approximate) Ramsey Properties Borsuk-Ulam Theorem (1) Case p = ∞ Bartosova, Lopez-Abad, Mbombo (2014), case p � = ∞ Ferenczi, Lopez-Abad, Mbombo and Todorcevic (2014). (1) The result for embeddings and d = 1 was proved by Odell, Schlumprecht and Rosenthal (1993), and by Matouˇ sek and R¨ odl (1995) independently. (2) The case p = 2 (i.e. the Hilbert case) is an indirect consequence of the fact that the Unitary group of ℓ 2 is extremely amenable , proved by Gromov and Milman (1983). (3) The result is true for real or complex Banach spaces. (4) There are several extensions to the context of operator spaces (Lupino and Lopez-Abad (2014)). J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 15 / 32
Applications. Extreme Amenability Extreme Amenability Definition Recall that a topological group G is extremely amenable (EA in short) when every (continuous) flow on a compact set K has a fixed point, that is, there is some p ∈ K such that g . p = p for every g ∈ G. The terminology is consistent with one of the characterizations of amenable groups: Every action by affine mappings on a compact convex set has a fixed point. J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 16 / 32
Applications. Extreme Amenability Extreme Amenability 1 The unitary group U ( ℓ 2 ), equipped with strong operator topology (Gromov-Milman, 1984). 2 Aut ( Q , ≤ ) the group of all order-preserving bijections of the rationals (Pestov, 1998). 3 In general automorphism groups of certain Fraiss´ e structures (Kechris-Pestov-Todorcevic). Namely Fraiss´ e class with structural Ramsey property. 4 Iso ( U ) where U is the universal Urysohn space. (Pestov, 2002) 5 The group Iso ( L p ( X , µ )) (for every 1 ≤ p < ∞ ) where ( X , µ ) is a standard Borel measure space with a non-atomic measure ( X , µ ), equipped with the strong operator topology. (Giordano and Pestov 2007) J. Lopez-Abad (ICMAT) Ramsey and EA FLORIANOPOLIS 17 / 32
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