Polarized partitions on the second level of the projective hierarchy (Q435206)

The authors study the question if for any set \(A\subseteq\omega^\omega\) from the projective class \(\Gamma\) one can find a sequence \((H_i)_{i\in\omega}\), where \(|H_i|\geq 2\), such that the set \(\prod_i H_i\) is included in \(A\) or disjoint with \(A\). If the answer is positive we say that sets from \(\Gamma\) have the polarized partition property. If in addition the \(H_i\) are bounded by some pre-defined recursive function we say that sets from \(\Gamma\) have the bounded polarized partition property. NEWLINENEWLINENEWLINE The question for analytic subsets has a positive answer (by a theorem of Di Prisco and Todorcevic). NEWLINENEWLINENEWLINE In the paper the authors study the cases \(\Gamma=\Sigma^1_2\) and \(\Gamma=\Delta^1_2.\) They compare the (bounded) polarized partition property for \(\Gamma\) with other regularity properties, for instance \(\forall a \exists x (x\) is eventually different over \(L[a])\). NEWLINENEWLINENEWLINE The authors examine the (bounded) polarized partition property for \(\Gamma\) in several standard models (Cohen model, Mathias model, \dots). They define a fat creature forcing which forces the bounded polarized property for \(\Sigma^1_2\) and does not add unbounded and splitting reals.