Large dilates of hypercube graphs in the plane

Abstract

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.