We show that if is an uncountable AD (almost disjoint) family of subsets of then the space does not admit a continuous selection; moreover, if is maximal then does not even admit a continuous selection on pairs, answering thus questions of T. Nogura.
We say that an ideal I on is semiproper if the corresponding poset is semiproper. In this paper we investigate properties of semiproper ideals on .
We give several topological/combinatorial conditions that, for a filter on ω, are equivalent to being a non-meager -filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager -filter. Here, we identify a filter with a subspace of through characteristic functions. Along the way, we generalize to non-meager -filters a result of Miller (1984) about -points, and we employ and give a new proof of results of Marciszewski (1998). We also employ a theorem...
A graph on is called -smooth if for each uncountable , is isomorphic to for some finite . We show that in various models of ZFC if a graph is -smooth, then is necessarily trivial, i.eėither complete or empty. On the other hand, we prove that the existence of a non-trivial, -smooth graph is also consistent with ZFC.
Chad, Knight & Suabedissen [Fund. Math. 203 (2009)] recently proved, assuming CH, that there is a 2-point set included in the union of countably many concentric circles. This result is obtained here without any additional set-theoretic hypotheses.
We give an equivalent, but simpler formulation of the axiom SEP, which was introduced in [9] in order to capture some of the combinatorial behaviour of models of set theory obtained by adding Cohen reals to a model of CH. Our formulation shows that many of the consequences of the weak Freese-Nation property of 𝒫(ω) studied in [6] already follow from SEP. We show that it is consistent that SEP holds while 𝒫(ω) fails to have the (ℵ₁,ℵ ₀)-ideal property introduced in [2]. This answers a question...
We prove a version of the Ramsey theorem for partitions of (increasing) n-tuples. We derive this result from a version of König's infinity lemma for ξ-large trees. Here ξ < ε₀ and the notion of largeness is in the sense of Hardy hierarchy.