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.
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The authors are grateful to Zoran Vondraček and the anonymous referee for several useful comments, which have improved the presentation.
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B. P. is supported by the Croatian Science Foundation.
The research leading to these results received funding from the Croatian Science Foundation under project UIP-2017-05-4129 (MUNHANAP).
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Kovač, V., Predojević, B. Large dilates of hypercube graphs in the plane. Anal Math (2024). https://doi.org/10.1007/s10476-024-00045-6
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DOI: https://doi.org/10.1007/s10476-024-00045-6