Set of vectors separated by at least a specified angle Suppose $\theta$ and $d$ are given.
How big can a set of $d$-dimensional vectors be such that no pair of them are at angle less than theta?
I particularly want an upper bound; that is, an $n=n(\theta,d)$ such that given $n$ $d$-dimensional vectors, there must be at least $2$ with angle less than theta between them.
Of course, the question can be rewritten in all sorts of ways, for example, coverings of the surface of the d-dimensional sphere by $(d-1)$-dimensional caps of given radius etc. 
The bound doesn't need to be tight. Something out by a factor of $(constant)^d$ might be fine
(although something more exact would be interesting too).
 A: One way of obtaining a lower bound is to apply the Johnson-Lindenstrauss lemma to an orthonormal basis.  This gives exponentially many vectors such that the angles between all pairs are arbitrarily close to $\pi/2$.
http://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma
A: 
This is a standard construction based on the fact that maximal $\theta$-packing is also an $\theta$-net.

Fix $d$.
Given $\theta>0$, consider $\theta$-packing of $S^d$; i.e. a set of $n=n(\theta)$ points $x_1,x_2,\dots,x_n$ in $S^d$ such that   $|x_ix_j|>\theta$ (we measure intrinsic distances in the sphere). 
Note that 


*

*$B(\tfrac\theta2,x_i)\cap B(\tfrac\theta2,x_i)=\varnothing$ for $i\not=j$ and 

*$\bigcup\limits_i B(\theta,x_i)=S^d$ (i.e. $\{x_1,x_2,\dots,x_n\}$ form a $\theta$-net in $S^d$)
Set $v(r)=\mathop{\rm vol}\{B(r,x)\subset S^d\}$.
Then $$n \cdot v(\tfrac\theta2) < \mathop{\rm vol}S^d < n\cdot v(\theta).$$
Clearly  $1\le \tfrac{v(\theta)}{v(\theta/2)}\le 2^d$.
Thus, $\mathop{\rm vol}S^d/v(\theta)$ gives $n$ up to factor $2^d$.
A: The subject name you are looking for is spherical codes. A good reference for this subject is Conway and Sloane's "Sphere Packings, Lattices, and Groups." In chapter 9 they give the details of the proof for the best bounds (I believe it is due to Levenstein, but don't have the book with me).
This ends up being related to density of sphere packings. There's a very elegant proof in the book which relates the answer to your question in dimension $n+1$ to the maximal density of sphere packing in dimension $n.$
Sorry I don't have my references with me, but this is all in chapter 9 of the book.
