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 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?

 A: Amit:
If ${\mathbb P}$ is a non-trivial separative partial order, and it is countable, an easy argument (back-and-forth) shows that it is forcing isomorphic to Cohen forcing. (This is an exercise in Chapter VII of Kunen's book, I believe, and it can be found in a few other sources, such as the appropriate chapter of the "Handbook of Boolean Algebras".)
It follows that if ${\mathbb P}$ is non-trivial, separative, and admits a countable dense subset, then ${\mathbb P}$ is again Cohen forcing. Now, ${\mathbb P}$ embeds into its Boolean completion which must then coincide with the Boolean completion of Cohen (Borel sets/meager) and therefore has size ${\mathfrak c}=|{\mathbb R}|$. This gives ${\mathfrak c}$ as an upper bound for $|{\mathbb P}|$.
Here is a cute application: Fix $\epsilon>0$, and let ${\mathbb P}^\epsilon$ be the collection of open subsets $p$ of ${\mathbb R}$ that are a finite union of intervals with rational end-points and such that the sum of their lengths (the Lebesgue measure of $p$) is less than $\epsilon$. The generic object gives us an open set that covers the ground model reals, and has measure $\epsilon$. Since ${\mathbb P}^\epsilon$ is countable, it is Cohen forcing. Since $\epsilon$ is arbitrary, it follows that adding a Cohen real makes the set of ground model reals have measure zero.

Edit: As Andreas Blass pointed out, one can actually prove the upper bound $|{\mathbb P}|\le{\mathfrak c}$ directly. This actually gives us a way of proving the characterization of Cohen forcing.
A: Here's an answer due to Andy Voellmer: If $\mathbb{P}$ has a dense subset $\{ p_i : i < \kappa \}$ of size at most $\kappa$ then by separativity, the map $f : \mathbb{P} \to 2 ^{\kappa}$ defined by $f(p) = \{ i < \kappa : p_i \leq p \}$ is an injection.  This is probably what Andreas Blass meant by "arguing directly from separativity."
