Abstract
We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matoušek. 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.
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