Charles University in Prague, Faculty of Mathematics and Physics
Abstract
summary: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=y}, and dist(f)=∥f∥Lip.∥f−1∥Lip (the {\sl 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