We prove new upper bound theorems on the consistency strengths of SPFA(θ), SPFA(θ-linked) and SPFA(θ+-cc). Our results are in terms of (θ,Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(𝔠-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper. As a corollary, SPFA(𝔠-linked) and PFA(𝔠-linked) are each equiconsistent with the existence of a Σ21-indescribable cardinal. Our upper bound for SPFA(𝔠-c.c.) is a Σ22-indescribable cardinal, which is consistent with V=L. Our upper bound for SPFA(𝔠+-linked) is a cardinal κ that is (κ+, Σ21)-subcompact, which is strictly weaker than κ+-supercompact. The axiom MM(𝔠) is a consequence of SPFA(𝔠+-linked) by a slight refinement of a theorem of Shelah. Our upper bound for SPFA(𝔠++-c.c.) is a cardinal κ that is (κ+, Σ22)-subcompact, which is also strictly weaker than κ+-supercompact.
"Hierarchies of forcing axioms I." J. Symbolic Logic 73 (1) 343 - 362, March 2008. https://doi.org/10.2178/jsl/1208358756