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.



large ccc forcing that preserves CH
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.


