# Optimal proofs of determinacy II

Itay Neeman

Abstract:

We present a general lemma which allows proving determinacy

from Woodin cardinals. The lemma can be used in many different

settings. As a particular application we prove the determinacy

of sets in $\Game^{(n)}(<\omega^2-\Pi^1_1)$, $n\geq 1$. The

assumption we use to prove $\Game^{(n)}(<\omega^2-\Pi^1_1)$

determinacy is optimal in the base theory of $\mbox{\sf ZFC} + \mbox{${\mathbf\Pi}^1_n$determinacy}$.