The Lipschitz geometry of segments of the infinite Hamming cube is studied. Tight estimates on the distortion necessary to embed the segments into spaces of continuous functions on countable compact metric spaces are given. As an application, the fir
st nontrivial lower bounds on the $C(K)$-distortion of important classes of separable Banach spaces, where $K$ is a countable compact space in the family $ { [0,omega],[0,omegacdot 2],dots, [0,omega^2], dots, [0,omega^kcdot n],dots,[0,omega^omega]} ,$ are obtained.
For every $alpha<omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $omega^{alphaomega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $omega^{alphaomega}$. In order to
prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates.
