  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.
