Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Embedding graphs in Euclidean space   The dimension of a graph $G$ is the smallest $d$ for which its vertices can
be embedded in $d$-dimensional Euclidean space in the sense that the distances
between endpoints of edges equal $1$ (but there may be other unit distances).
Answering a question of Erd\H{o}s and Simonovits [Ars Combin. 9 (1980)
229--246], we show that any graph with less than $\binom{d+2}{2}$ edges has
dimension at most $d$. Improving their result, we prove that that the dimension
of a graph with maximum degree $d$ is at most $d$. We show the following Ramsey
result: if each edge of the complete graph on $2d$ vertices is coloured red or
blue, then either the red graph or the blue graph can be embedded in Euclidean
$d$-space. We also derive analogous results for embeddings of graphs into the
$(d-1)$-dimensional sphere of radius $1/\sqrt{2}$.

6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
1
The categories of this paper are: cs
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph