Low distortion embeddings of some metric graphs into Banach spaces

Abstract

We give a simple example of a countable metric graph M such that M Lipschitz embeds with distortion strictly less than 2 into a Banach space X only if X contains an isomorphic copy of l ₁. Further we show that, for each ordinal α < ω ₁, the space C([0, ω ^α]) does not Lipschitz embed into C(K) with distortion strictly less than 2 unless K ^(α) ≠ 0. Also $C\left( {\left[ {0,{\omega ^{{\omega ^\alpha }}}} \right]} \right)$ does not Lipschitz embed into a Banach space X with distortion strictly less than 2 unless Sz(X) ≥ ω ^α+1.