Polarized partitions on the second level of the projective hierarchy

A subset A of the Baire space ω 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. DiPrisco and Todorčević [6] proved that both partition properties are true for analytic sets A. In this paper we investigate these properties on the ∆2and Σ 1 2-levels of the projective hierarchy, i.e., we investigate the strength of the statements “all ∆2/Σ 1 2 sets satisfy the (bounded) polarized partition property” and compare it to similar statements involving other well-known regularity properties.