Large dilates of hypercube graphs in the plane

TL;DR

This paper proves that large dilates of hypercube graphs can be embedded in any measurable subset of the plane with positive upper Banach density, extending known results beyond trees.

Contribution

It demonstrates the first dimensionally sharp embedding of hypercube graphs in positive density sets in the plane, beyond previously known tree structures.

Findings

Large dilates of hypercube graphs are contained in positive density sets

First examples of non-tree distance graphs with sharp embedding results

Extends embedding theory to higher-dimensional hypercube structures

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.