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