We say that a coloring ${c: [\kappa]^n\to 2}$ is continuous if it is continuous with respect to some second countable topology on κ. A coloring c is potentially continuous if it is continuous in some ${\aleph_1}$ -preserving extension of the set-theoretic universe. Given an arbitrary coloring ${c:[\kappa]^n\to 2}$ , we define a forcing notion ${\mathbb P_c}$ that forces c to be continuous. However, this forcing might collapse cardinals. It turns out that ${\mathbb P_c}$ is c.c.c. if and only if c (...) is potentially continuous. This gives a combinatorial characterization of potential continuity. On the other hand, we show that adding ${\aleph_1}$ Cohen reals to any model of set theory introduces a coloring ${c:[\aleph_1]^2 \to 2}$ which is potentially continuous but not continuous. ${\aleph_1}$ has no uncountable c-homogeneous subset in the Cohen extension, but such a set can be introduced by forcing. The potential continuity of c can be destroyed by some c.c.c. forcing. (shrink)

A Boolean algebra B is said to be openly generated if {A: A ≤rc B, |A| = ℵ0} includes a club subset of [ B]ℵ0 . We show: (V = L). For any cardinal κ there exists an L∞κ-free Boolean algebra which is not openly generated (Proposition 4.1). (MA+(σ-closed)). Every L∞ℵa -free Boolean algebra is openly generated (Theorem 4.2). The last assertion follows from a characterization of openly generated Boolean algebras under MA+(σ-closed) (Theorem 3.1). Using this characterization we also prove (...) the independence of problem 7 in Scepin [15] (Proposition 4.3 and Theorem 4.4). (shrink)