Large dilates of hypercube graphs in the plane

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