I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing. I am mainly interested in proofs of combinatorial properties of complete Boolean algebras such as the many cardinal invariants on Boolean algebras. The proofs of results that I have in mind have the following form.

$\bullet$ One translates combinatorial properties of complete Boolean algebras $B$ to properties of the Boolean valued models $V^{B}$.

$\bullet$ One proves things about the Boolean valued models $V^{B}$.

$\bullet$ One translates the results about the Boolean valued models $V^{B}$ back to results about the complete Boolean algebras $B$.

I am interested in proofs of results where the only known proof uses forcing or where the proof without forcing is more difficult or at least equally difficult.

This question and this question ask about ZFC results that can be proven using forcing, but now I am only asking about combinatorial results about complete Boolean algebras proven using forcing using the strategy mentioned above.

I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing. I am mainly interested in proofs of combinatorial properties of complete Boolean algebras such as the many cardinal invariants on Boolean algebras. The proofs of results that I have in mind have the following form.

$\bullet$ One translates combinatorial properties of complete Boolean algebras $B$ to properties of the Boolean valued models $V^{B}$.

$\bullet$ One proves things about the Boolean valued models $V^{B}$.

$\bullet$ One translates the results about the Boolean valued models $V^{B}$ back to results about the complete Boolean algebras $B$.

I am interested in proofs of results where the only known proof uses forcing or where the proof without forcing is more difficult or at least equally difficult.

This question and this question ask about ZFC results that can be proven using forcing, but now I am only asking about combinatorial results about complete Boolean algebras proven using forcing using the strategy mentioned above.

I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing. The proofs of results that I have in mind have the following form.

$\bullet$ One translates Boolean algebraic properties of complete Boolean algebras $B$ to properties of the Boolean valued models $V^{B}$.

$\bullet$ One proves things about the Boolean valued models $V^{B}$.

$\bullet$ One translates the results about the Boolean valued models $V^{B}$ back to results about the complete Boolean algebras $B$.

I am interested in proofs of results where the only known proof uses forcing or where the proof without forcing is more difficult or at least equally difficult.

This question and this question ask about ZFC results that can be proven using forcing, but now I am only asking about results about complete Boolean algebras proven using forcing using the strategy mentioned above.

I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing. I am mainly interested in proofs of combinatorial properties of complete Boolean algebras such as the many cardinal invariants on Boolean algebras. The proofs of results that I have in mind have the following form.

$\bullet$ One translates combinatorial properties of complete Boolean algebras $B$ to properties of the Boolean valued models $V^{B}$.

$\bullet$ One proves things about the Boolean valued models $V^{B}$.

$\bullet$ One translates the results about the Boolean valued models $V^{B}$ back to results about the complete Boolean algebras $B$.

I am interested in proofs of results where the only known proof uses forcing or where the proof without forcing is more difficult or at least equally difficult.

This question and this question ask about ZFC results that can be proven using forcing, but now I am only asking about combinatorial results about complete Boolean algebras proven using forcing using the strategy mentioned above.