I am looking for a reference for the (classical) one way randomised communication complexity of disjointness when the universe can be large. Say Alice and Bob both have sets of size $m$ chosen from a universe of size $U$ and Bob wants to determine if the intersection of their sets is empty or not. I would like prob of error $<1/3$, say.
I can find the standard $\Omega(m)$ bit lower bound and some work on two-way communication complexity, but is there a reference for something tighter for one-way?
EDIT: I should have specified that I am interested in the private randomness (not public coin) model.