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.

17 pages; v2: corrected typos, minor changes in the exposition