Partial strong compactness and squares

We analyze the connection between some properties of partially strongly compact cardinals: the completion of filters of certain size and instances of the compactness of Lκ,κ. Using this equivalence we show that if any κ-complete filter on λ can be extended to a κ-complete ultrafilter and λ ^<κ = λ then (µ) fails for all regular µ ∈ [κ, 2 ^λ]. As an application, we improve the lower bound for the consistency strength of κ-compactness, a case which was explicitly considered by Mitchell.