Using a game characterization of distributivity, we show that base matrices for P(ω)/fin of regular height larger than h necessarily have maximal branches which are not cofinal.

We construct a model in which there exists a distributivity matrix of regular height λ larger than h; both λ = c and λ < c are possible. A distributivity matrix is a refining system of mad families… Expand

A classical theorem of Balcar, Pelant, and Simon says that there is a base matrix of height h, where h is the distributivity number of P(ω)/fin. We show that if the continuum c is regular, then there… Expand

A model of ZFC is constructed in which the distributivity cardinal h is , and in which there are no ω 2 -towers in [ ω ] ω so that any base-matrix tree in this model has no cofinal branches.Expand

A partial converse to Jech's theorem is established, namely it gives cardinality conditions on ℬ under which II having a winning strategy implies ω -closure, and considers games of length > ω and generalize Jech’s theorem to these games.Expand

MR. McCOLL still expresses surprise at my declining to answer a Yes or No question which he was pleased to put to me in NATURE (vol. xxiv. p. 124). It was, I should think, almost unique in a… Expand

It is shown that every ρ-closed partial order of size continuum has a base tree and that σ-closed forcing notions of density 𝔠 correspond exactly to regular suborders of the collapsing algebra Coll(ω1, 2ω).Expand