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).