Ramsey-like properties for bi-Lipschitz mappings of finite metric spaces

Let (X, ρ), (Y, σ) be metric spaces and f : X → Y an injective mapping. We put ‖f‖Lip = sup{σ(f(x), f(y))/ρ(x, y); x, y ∈ X, x 6= y}, and dist(f) = ‖f‖Lip .‖f ‖Lip (the distortion of the mapping f). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let X be a finite metric space, and let ε > 0, K be given numbers. Then there exists a finite metric space Y , such that for every mapping f : Y → Z (Z arbitrary metric space) with dist(f) < K one can find a mapping g : X → Y , such that both the mappings g and f |g(X) have distortion at most (1 + ε). If X is isometrically embeddable into a lp space (for some p ∈ [1,∞]), then also Y can be chosen with this property.