  We consider strong combinatorial principles for sigma-directed families of
countable sets in the ordering by inclusion modulo finite, e.g. P-ideals of
countable sets. We try for principles as strong as possible while remaining
compatible with CH, and we also consider principles compatible with the
existence of nonspecial Aronszajn trees. The main thrust is towards abstract
principles with game theoretic formulations. Some of these principles are
purely combinatorial, while the ultimate principles are primarily combinatorial
but also have aspects of forcing axioms.
