Abstract
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.
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The research for this paper was supported by G.I.F. Research Grant No. I-802-195.6/2003. The author would like to thank Uri Abraham for very fruitful discussions on the subject of this article.
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Geschke, S. Potential continuity of colorings. Arch. Math. Logic 47, 567–578 (2008). https://doi.org/10.1007/s00153-008-0097-z
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DOI: https://doi.org/10.1007/s00153-008-0097-z