### Analytic Colorings

by Kubis and Shelah. [KbSh:802]

Annals Pure and Applied Logic, 2003

We investigate the existence of perfect homogeneous sets for
analytic colorings. An analytic coloring of X is an analytic
subset of [X]^N, where N>1 is a natural number. We define an
absolute rank function on trees representing analytic colorings,
which gives an upper bound for possible cardinalities of homogeneous
sets and which decides whether there exists a perfect homogeneous
set. We construct universal sigma-compact colorings of any
prescribed rank gamma < omega_1 . These colorings consistently
contain homogeneous sets of cardinality aleph_gamma but they do
not contain perfect homogeneous sets. As an application, we discuss
the so-called defectedness coloring of subsets of Polish linear
spaces.

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