Summary: After presenting a general setting in which to look at forcing axioms, we give a hierarchy of generalized bounded forcing axioms that correspond level by level, in consistency strength, with the members of a natural hierarchy of large cardinals below a Mahlo. We give a general construction of models of generalized bounded forcing axioms. Then we consider the bounded forcing axiom for a class of partially ordered sets \(\Gamma_1\) such that, letting \(\Gamma_0\) be the class of all stationary-set-preserving partially ordered sets, one can prove the following:
(a) \(\Gamma_0\subseteq \Gamma_1\);
(b) \(\Gamma_0= \Gamma_1\) if and only if \(NS_{\omega_1}\) is \(\aleph_1\)-dense;
(c) If \(P\notin\Gamma_1\), then BFA\((\{P\})\) fails.
We call the bounded forcing axiom for \(\Gamma_1\) Maximal Bounded Forcing Axiom (MBFA). Finally we prove MBFA, consistent relative to the consistency of an inaccessible \(\Sigma_2\)-correct cardinal which is a limit of strongly compact cardinals.