Lipschitz representations of subsets of the cube

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We show that for any class of uniformly bounded functions H H with a reasonable combinatorial dimension, the vast majority of small subsets of the n n -dimensional combinatorial cube cannot be represented as a Lipschitz image of a subset of H H , unless the Lipschitz constant is very large. We apply this result to the case when H H consists of linear functionals of norm at most one on a Hilbert space.