# Article

 Title: Ramsey-like properties for bi-Lipschitz mappings of finite metric spaces (English) Author: Matoušek, Jiří Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 33 Issue: 3 Year: 1992 Pages: 451-463 . Category: math . Summary: Let $(X,\rho)$, $(Y,\sigma)$ be metric spaces and $f:X\to Y$ an injective mapping. We put $\|f\|_{Lip} = \sup \{\sigma (f(x),f(y))/\rho(x,y)$; $x,y\in X$, $x\neq y\}$, and $\operatorname{dist}(f)= \|f\|_{Lip}.\|f^{-1}\|_{Lip}$ (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 $\varepsilon>0$, $K$ be given numbers. Then there exists a finite metric space $Y$, such that for every mapping $f:Y\to Z$ ($Z$ arbitrary metric space) with \$\operatorname{dist}(f)

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