In general, two overlapping Boolean algebras always extend to a common Boolean algebra, but three may not. We prove a new sufficient condition for $n$ overlapping Boolean algebras to have a common extension. Combining this with the set-theoretic technique of long $\omega_1$-approximation sequences (also known as Davies sequences), we obtain a flexible method of constructing (in ZFC) arbitrarily large Boolean algebras as direct limits of countable Boolean algebras. Along the way, we develop some… CONTINUE READING