  We give the first lower bound on the $\scr C(K)$-distortion of the class of
separable Banach spaces, for $K$ a countable compact in the family $\{
[0,\omega],[0,\omega\cdot2],\cdots, [0,\omega^2], \cdots, [0,\omega^k\cdot
n],\cdots,[0,\omega^\omega]\}$.
