Homogeneity for open partitions of pairs of reals

Abstract

We prove a partition theorem for analytic sets of reals, namely, if A ⊆ R A \subseteq \mathbb {R} is analytic and [ A ] 2 = K 0 ∪ K 1 {[A]^2} = {K_0} \cup {K_1} with K 0 {K_0} relatively open, then either there is a perfect 0 0 -homogeneous subset or A A is a countable union of 1 1 -homogeneous subsets. We also show that such a partition property for coanalytic sets is the same as that each uncountable coanalytic set contains a perfect subset. A two person game for this partition property is also studied. There are some applications of such partition properties.