Coherent sequences and threads

Abstract The combinatorial principle □ ( λ ) says that there is a coherent sequence of length λ that cannot be threaded. If λ = κ + , then the related principle □ κ implies □ ( λ ) . Let κ ⩾ ℵ 2 and X ⊆ κ . Assume both □ ( κ ) and □ κ fail. Then there is an inner model N with a proper class of strong cardinals such that X ∈ N . If, in addition, κ ⩾ 2 ℵ 0 and n ω , then there is an inner model M n ( X ) with n Woodin cardinals such that X ∈ M n ( X ) . In particular, by Martin and Steel, Projective Determinacy holds. As a corollary to this and results of Todorcevic and Velickovic, the Proper Forcing Axiom for posets of cardinality ( 2 ℵ 0 ) + implies Projective Determinacy.