Large dilates of hypercube graphs in the plane
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917821985849344 |
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| author | Kovač, Vjekoslav Predojević, Bruno |
| author_facet | Kovač, Vjekoslav Predojević, Bruno |
| contents | We study a distance graph $Γ_n$ 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 $Γ_n$. This provides the first examples of distance graphs other than the trees for which a dimensionally sharp embedding in positive density sets is known. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_14791 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Large dilates of hypercube graphs in the plane Kovač, Vjekoslav Predojević, Bruno Classical Analysis and ODEs Combinatorics We study a distance graph $Γ_n$ 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 $Γ_n$. This provides the first examples of distance graphs other than the trees for which a dimensionally sharp embedding in positive density sets is known. |
| title | Large dilates of hypercube graphs in the plane |
| topic | Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2309.14791 |