On Box, Weak Box and Strong Compactness

One of the most important goals of set theorists over the last few years has been to re-prove old results which previously had used very strong assumptions from hypotheses which, at least prima facie, are weaker. Examples of these abound, including, but certainly not limited to, the work of Woodin and Cummings (see [3]) on the Singular Cardinals Problem, in which results previously obtained by Magidor[5, 6] using supercompactness and hugeness were re-proven using hypermeasurability. This paper continues the work of [1] along these lines, re-proving a result of Ben-David and Magidor[2] using strong compactness instead of supercompactness. In [2], the authors show that Con (ZFC + GCH + 3K[K is K supercompact]) => Con (ZFC +•«„, + "' DKJ, where • * and • * are, respectively, the combinatorial principles box and weak box at the cardinal K. (See [2] for the appropriate definitions.) We prove the following.