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We study the modulus of dual families of $k$- and $(n-k)$-dimensional Lipschitz chains of Euclidean $n$-cubes and establish half of the modulus duality identity.

We prove that there exists a constant $\epsilon> 0$ with the following property: if $K \subset {\mathbb {R}}^2$ is a compact set that contains no pair of the form $\{x, x + (z, z^{2})\}$ for $z \neq 0$, then $\dim _{\textrm {H}} K \leq 2 - \epsilon $.