Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Embedding Euclidean Distance Graphs in $\mathbb{R}^n$ and $\mathbb{Q}^n$   For $S \subseteq \mathbb{R}$, positive integer $n$, and $d > 0$, let $G(S^n,
d)$ be the graph whose vertex set is $S^n$ where any two vertices are adjacent
if and only if they are Euclidean distance $d$ apart. The primary question we
will consider in our work is as follows. Given $n$ and distance $d$ actually
realized as a distance between points of the rational space $\mathbb{Q}^n$,
does there exist a finite graph $G$ that appears as a subgraph of
$G(\mathbb{Q}^n, d)$ but not as a subgraph of $G(\mathbb{R}^{n-1}, 1)$? We
answer this question affirmatively for $n \leq 5$, and along the way, resolve a
few related questions as well.

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The categories of this paper are: cs
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The categories of this paper are: physics
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom