  We study a distance graph $\Gamma_n$ that is isomorphic to the $1$-skeleton
of an $n$-dimensional unit hypercube. We show that every measurable set of
positive upper Banach density in the plane contains all sufficiently large
dilates of $\Gamma_n$. This provides the first examples of distance graphs
other than the trees for which a dimensionally sharp embedding in positive
density sets is known.
