Abstract

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 \.