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).