Given a set of points on the plane you can construct unit distance graph by taking the points as vertices, and setting two points as adjacent if the distance between the two points is one.

I am curious about something analogous to this on a sphere.

Is there anything known about, for a fixed angle $\theta$ and dimension $n$, the graph whose vertices are the unit vectors in $\mathbb{R}^n$ with a pair of vertices adjacent if only if the angle between the two vectors are exactly $\theta$. I will refer to this as $G(n, \theta)$.

I am wondering if there are any bounds on the chromatic number of $G(n, \theta)$? I know for example that it contains $K_{n-1}$ because, if you set $\alpha = \sqrt{\cos{\theta}}$ and $\beta = \sqrt{1-\cos{\theta}}$ the vectors

- $(\alpha, \beta, 0, 0,\ldots, 0)$
- $(\alpha, 0, \beta, 0, \ldots, 0)$
- ...
- $(\alpha, 0, 0, \ldots, \beta)$

are all adjacent.

If you set $n = N^2 + N + 1$ and $\theta = \arccos{\frac{1}{N+1}}$, then you can get $K_n$ by having the dimensions correspond to the points in a projective plane of size $n$, and having the unit vectors correspond to the $n$ lines with $N+1$ non-zero entries each $\frac{1}{\sqrt{N+1}}$ for each point in the line.

I there a way to get subgraphs in $G(n, \theta)$ with high chromatic numbers?