2010 seminar talk: Basis theorems for continuous colorings
Talk held by Stefan Geschke (Hausdorff Center for Mathematics, Universität Bonn, Germany) at the KGRC seminar on 2010-12-16.
The collection of n-element subsets of a Hausdorff space carries a natural topology. A continuous n-coloring on a Polish space X is a continuous map that assigns to each n-element subset of X one of two colors.
An n-coloring is uncountably homogeneous if the underlying space X is not the union of countably many sets on which the coloring is constant.
Generalizing a previous result about 2-colorings (i.e., graphs) and answering a question of Ben Miller, it is shown that the class of uncountably homogeneous, continuous n-colorings on Polish spaces has a finite basis.
I.e., there is a finite collection of uncountably homogeneous, continuous n-colorings on the Cantor space such that every uncountably homogeneous, continuous n-coloring on any Polish space contains a copy of one of the finitely many colorings.
This complements some recent results of Lecomte and Miller on the nonexistence of small bases for uncountably chromatic analytic graphs.