Analytic colorings

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 σ-compact colorings of any prescribed rank γ<ω1. These colorings consistently contain homogeneous sets of cardinality γ but they do not contain perfect homogeneous sets. As an application, we discuss the so-called defectedness coloring of subsets of Polish linear spaces