Abstract   We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matousek. Then we show that for every   K  >  1   every uncountable Polish space has a perfect subset that   K  -bi-Lipschitz embeds into the real line. Finally we study decompositions of infinite separable metric spaces into subsets that, for some   K  >  1  ,   K  -bi-Lipschitz embed into the real line.
['Stefan Geschke']
Low-Distortion Embeddings of Infinite Metric Spaces into the Real Line