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)

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ 1 many maximal antichains of a proper forcing notion, each of size ℵ 1 , there is a directed set meeting all these antichains. A regular cardinal κ is called Σ 1 -reflecting, if for any regular cardinal χ, for all formulas $\varphi, "H(\chi) \models`\varphi'"$ implies " $\exists\delta . We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the bounded (...) proper forcing axiom is exactly the existence of a Σ 1 -reflecting cardinal (which is less than the existence of a Mahlo cardinal). We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure. (shrink)