# 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.

### Abstract

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.