2 added 158 characters in body

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 what about consider the following analogous questions:

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

and

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?

1

# A problem about posets similar to Suslin's problem

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 what about the following analogous questions:

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

and

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?