平面超立方体图的大扩张

Pub Date : 2024-09-15 DOI:10.1007/s10476-024-00045-6

V. Kovač, B. Predojević

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引用次数: 0

摘要

我们研究了一个距离图（\γ_n\），它与（n\）维单位超立方体的（1\）骨架同构。我们证明了平面上每一个正上巴纳赫密度的可测集合都包含了 \(\Gamma_n\) 的所有足够大的扩张。这提供了除树以外的距离图的第一个例子，已知这些距离图在正密度集合中的维度锐嵌入。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Large dilates of hypercube graphs in the plane

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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