Analytic colorings

Wiesław Kubiś, Saharon Shelah
2003 Annals of Pure and Applied Logic  
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 deÿne 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 -compact colorings of any prescribed rank ¡ !1. These colorings consistently contain
more » ... omogeneous sets of cardinality ℵ but they do not contain perfect homogeneous sets. As an application, we discuss the so-called defectedness coloring of subsets of Polish linear spaces.
doi:10.1016/s0168-0072(02)00110-0 fatcat:slfxympz7nhhtipoooeotevo6e