Recall that an inaccessible cardinal $\kappa$ is a Woodin cardinal if for every $A\subseteq V_\kappa$ there is an unbounded set in $\kappa$ of $\lambda$ such that $V_\kappa\models\lambda$ is $A$-strong.
This raises the question, of whether or not intermediate notions have been defined in the literature, and what sort of results can they be used to obtain?
For example, $\Sigma^n_m$-Woodin is an inaccessible cardinal $\kappa$ such that for every $A$ which is $\Sigma^n_m$-definable over $V_\kappa$, there is an unbounded set of $\lambda<\kappa$ such that $V_\kappa\models\lambda$ is $A$-strong.
It seems probable that there is such hierarchy of definable Woodin-ness, and that it might be used to obtain partial determinacy (and saturation) results.
Do these notions exist in the literature, and if so do they have interesting consequences?