Potential Continuity of Colorings
We say that a coloring c: [κ]n→ 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 ℵ1-preserving extension of the set-theoretic universe. Given an arbitrary coloring c: [κ]n→ 2, we define a forcing notion Ρc that forces c to be continuous. However, this forcing might collapse cardinals. It turns out that Pc 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 ℵ1 Cohen reals to any model of set theory introduces a coloring c: [ℵ1]2→ 2 which is potentially continuous but not continuous. ℵ1 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.
Geschke, Stefan. (2008). "Potential Continuity of Colorings". Archive for Mathematical Logic, 47(6), 567-578.