TITLE: large ccc forcing that preserves CH
QUESTION [9 upvotes]: Can you name a ccc forcing with the following properties?
1) Atomless and separative
2) The least size of a dense set is large, say at least $\aleph_3$, hopefully as big as you like.
3) Existence is consistent with CH.
4) Preserves CH.
If you can think of any, please list as many as you can.

REPLY [6 votes]: Thanks to Michael Blackmon for this idea:
If P has the ccc and preserves CH, then there is a contiuum sized regular suborder R that adds all the reals P will add.  The factor forcing P/R is ccc and $\omega$-distributive, a Suslin algebra.  A theorem (of Jech?) says that any Suslin algebra has size at most $2^{\omega_1}$.  So there is a bound on the size of P.