Potentially continuous colorings
by
Stefan Geschke
Free University of Berlin

A coloring of the n-element subsets of a given uncountable set is continuous if it is continuous with respect to a second countable topology on the set. A coloring is potentially continuous if it can be forced to be continuous without collapsing the first uncountable cardinal. We show that every potentially continuous coloring can be forced to be continuous by a canonical c.c.c. forcing. This gives a combinatorial characterization of potentially continuous colorings. We also give an example of a potentially continuous coloring that is not continuous.

Date received: March 20, 2006