COMBINED MAXIMALITY PRINCIPLES UP TO LARGE CARDINALS

The motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals kappa (even only allowing kappa itself as a parameter in the Maximality Principle for <kappa-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal kappa (instead of infinity), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high.