Polarized partitions on the second level of the projective hierarchy

A subset A of the Baire space ωω satisfies the polarized partition property if there is an infinite sequence 〈Hi∣i∈ω〉 of finite subsets of ω, with |Hi|≥2, such that ∏iHi⊆A or ∏iHi∩A=∅. It satisfies the bounded polarized partition property if, in addition, the Hi are bounded by some pre-determined recursive function. Di Prisco and Todorčević (2003) [6] proved that both partition properties are true for analytic sets A. In this paper we investigate these properties on the Δ21- and Σ21-levels of the projective hierarchy, i.e., we investigate the strength of the statements “all Δ21/Σ21 sets satisfy the (bounded) polarized partition property” and compare it to similar statements involving other well-known regularity properties.