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On metric Ramsey-type phenomena. (English) Zbl 1114.46007
This paper deals with extremely interesting nonlinear analogues of Dvoretzky’s Theorem (originally for finite-dimensional Banach spaces) in the context of metric spaces, with a point of view that may also let us see the results as part of combinatorial Ramsey theory: given a finite metric space on $$n$$ points, the problem is to identify the subspace of largest cardinality which can be embedded with a given (metric) distortion in Hilbert space. Hilbert space plays either the central role it has in Banach space theory, or can be seen as a “highly-structured” substructure in the spirit of Ramsey theory, the latter being, as the authors put it, the study of how “large systems necessarily contain large, highly structured sub-systems”.
This paper is long, deep and highly technical, so let us merely state here what the authors themselves announce in their introduction: they provide nearly tight upper and lower bounds on the cardinality of the “Hilbertian” subspaces in terms of $$n$$ and the desired distortion. The main theorem states that for any $$\varepsilon>0$$, every $$n$$ point metric space contains a subset of size at least $$n^{1-\varepsilon}$$ which is embeddable in Hilbert space with $$O(\log(1/\varepsilon)/\varepsilon)$$ distortion. The bound on the distortion is tight up to the $$\log(1/\varepsilon)$$ factor.

##### MSC:
 46B07 Local theory of Banach spaces 05D10 Ramsey theory 54E35 Metric spaces, metrizability
##### Keywords:
metric space; Ramsey theory; Dvoretzky’s theorem; distortion
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