Several things are called Bollobas's theorem. Try the following: http://iti.mff.cuni.cz/series/files/iti305.pdf
The proof of Bollobas's theorem on subspaces uses the fact that we can factor out by a subspace in general position. This requires that the Field is infinite. How can I prove the finite field version of the theorem?
Hmmm...theorem 1.1 is a purely combinatoric one , no fields or vector spaces whatsoever mentioned. perhaps you mean theorem 1.3, or maybe 1.4? If 1.3 then, as it's written there, Lovasz proved it and a reference to one of his papers is at the end of the file (and pay attention that this theorem is for "arbitrary fields"), and if you meant 1.4 then it is proved in that paper and it's for FINITE fields....I think this all pretty much covers what you wanted, doesn't it?!