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.
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