Low distortion embeddings between $C(K)$ spaces
  We show that, for each ordinal $\alpha<\omega_1$, the space $C([0,\omega^\alpha])$ does not embed into $C(K)$ with distortion strictly less than $2$ unless $K^{(\alpha)}\neq \emptyset$. 
Anton\'in Proch\'azka and Luis S\'anchez-Gonz\'alez
math.FA