Polarized partitions on the second level of the projective hierarchy
Jörg Brendle, Yurii Khomskii
Abstract:
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.