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

A topological obstruction for small-distortion embeddability into spaces of continuous functions on countable compact metric spaces