What is the relation between the existence of a cryptographic hash function and the existence of a PRG?
There is a black-box separation between one-way functions and collision resistant hash functions. This was proven at Eurocrypt 1998 by Dan Simon, in the paper entitled Finding collisions on a one-way street: Can secure hash functions be based on general assumptions?. Of course, this doesn't mean that it's not possible using non-black-box reductions, but no one knows how to do this.