PP-2011-09: Polarized partitions on the second level of the projective hierarchy

PP-2011-09: Brendle, Jörg and Khomskii, Yurii (2011) Polarized partitions on the second level of the projective hierarchy. [Report]

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A subset $A$ of the Baire space satisfies the "polarized partition property" if there is an infinite sequence $< H_i | i \in \omega >$ of finite subsets of $\omega$, with $|H_i| \geq 2$, such that $\prod_i H_i \subseteq A$ or $\prod_i H_i \cap A = \varnothing$. It satisfies the "bounded polarized partition property" if, in addition, the $H_i$ are bounded by some pre-determined recursive function. DiPrisco and Todorcevic proved that both partition properties are true for analytic sets. In this paper we investigate these properties on the $\Delta^1_2$- and $\Sigma^1_2$-levels of the projective hierarchy, i.e., we investigate the strength of the statements "all $\Delta^1_2$ / $\Sigma^1_2$ sets satisfy the (bounded) polarized partition property" and compare it to similar statements involving other well-known regularity properties.