It was proved recently that Telgársky’s conjecture, which concerns partial information strategies in the Banach–Mazur game, fails in models of \. The proof introduces a combinatorial principle that is shown to follow from \, namely: \::Every separative poset \ with the \-cc contains a dense sub-poset \ such that \ for every \. We prove this principle is independent of \ and \, in the sense that \ does not imply \, and \ does not imply \ assuming the consistency (...) of a huge cardinal. We also consider the more specific question of whether \ holds with \ equal to the weight-\ measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of \. (shrink)

It is consistent that there is a compact space of character less than the splitting number in which there are no converging sequences. Such a space is an Efimov space.

It is shown to be consistent that countable, Fréchet,α 1-spaces are first countable. The result is obtained by using a countable support iteration of proper partial orders of lengthω 2.

We construct a model in which the splitting number is large and every ultrafilter has a small subset with no pseudo-intersection.

Partitioner algebras are defined by Baumgartner and Weese 619) as a natural tool for studying the properties of maximal almost disjoint families of subsets of ω. We prove from PFA+ and that there exists a partitioner algebra which contains a subalgebra which is not representable as a partitioner algebra.