Suslin's problem is:

Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$?

The answer is that it's independent of ZFC. The related question:

Given a complete dense linear order without endpoints, if it's separable must it be isomorphic to $\mathbb{R}$?

has a positive answer under ZFC. Now consider the following analogous questions:

Given a non-trivial separative forcing poset, if it has the ccc must it have size at most continuum?

The answer to this is no, for example the Cohen forcing that adds more than continuum-many reals is ccc but has size greater than continuum. So what about:

Given a non-trivial separative forcing poset, if it's separable (i.e. has a countable dense set) must it have size at most continuum?

`$2^\kappa$`

. (Even my use of the regular open algebra seems to be overkill; you could argue directly from separativity.) $\endgroup$ – Andreas Blass Nov 20 '10 at 19:35