Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Orthogonal colorings of the sphere   An orthogonal coloring of the two-dimensional unit sphere $\mathbb{S}^2$, is
a partition of $\mathbb{S}^2$ into parts such that no part contains a pair of
orthogonal points, that is, a pair of points at spherical distance $\pi/2$
apart. It is a well-known result that an orthogonal coloring of $\mathbb{S}^2$
requires at least four parts, and orthogonal colorings with exactly four parts
can easily be constructed from a regular octahedron centered at the origin. An
intriguing question is whether or not every orthogonal 4-coloring of
$\mathbb{S}^2$ is such an octahedral coloring. In this paper we address this
question and show that if every color class has a non-empty interior, then the
coloring is octahedral. Some related results are also given.

14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci