The property of strong compactness may be weakened by only requiring this compactness property to hold when the original collection of statements has cardinality below a certain cardinal λ; we may then refer to λ-compactness.
A cardinal κ is weakly compact if and only if it is κ-compact; this was the original definition of that concept.
The consistency strength of strong compactness is strictly above that of a Woodin cardinal.
However, a proof is unlikely until a canonical inner model theory for supercompact cardinals is developed.
Jech obtained a variant of the tree property which holds for an inaccessible cardinal if and only if it is strongly compact.