#### Title

Potential Continuity of Colorings

#### Document Type

Article

#### Publication Date

9-1-2008

#### DOI

http://dx.doi.org/10.1007/s00153-008-0097-z

#### Abstract

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 ℵ

_{1}-preserving extension of the set-theoretic universe. Given an arbitrary coloring

*c*: [

*κ*]

^{n}→ 2, we define a forcing notion

**Ρ**

*that forces*

_{c }*c*to be continuous. However, this forcing might collapse cardinals. It turns out that

**P**

*is c.c.c. if and only if*

_{c}*c*is potentially continuous. This gives a combinatorial characterization of potential continuity. On the other hand, we show that adding ℵ

_{1}Cohen reals to any model of set theory introduces a coloring

*c*: [ℵ

_{1}]

^{2}→ 2 which is potentially continuous but not continuous. ℵ

_{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.

#### Publication Information

Geschke, Stefan. (2008). "Potential Continuity of Colorings". *Archive for Mathematical Logic,** 47*(6), 567-578.