Homogeneity for open partitions of pairs of reals

Feng, Qi

doi:10.1090/S0002-9947-1993-1113695-5

Homogeneity for open partitions of pairs of reals
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Trans. Amer. Math. Soc. 339 (1993), 659-684 Request permission

Abstract:

We prove a partition theorem for analytic sets of reals, namely, if $A \subseteq \mathbb {R}$ is analytic and ${[A]^2} = {K_0} \cup {K_1}$ with ${K_0}$ relatively open, then either there is a perfect $0$-homogeneous subset or $A$ is a countable union of $1$-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.

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