A new ⋄-like principle ${\diamond}_{}$ consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that $\neg {\diamond}_{}$ is consistent with CH and that in many models of = ω₁ the principle ${\diamond}_{}$ holds. As ${\diamond}_{}$ implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that ${\diamond}_{}$ holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁ in such models....

Rational numbers are used to classify maximal almost disjoint (MAD) families of subsets of the integers. Combinatorial characterization of indestructibility of MAD families by the likes of Cohen, Miller and Sacks forcings are presented. Using these it is shown that Sacks indestructible MAD family exists in ZFC and that $\U0001d51f=\U0001d520$ implies that there is a Cohen indestructible MAD family. It follows that a Cohen indestructible MAD family is in fact indestructible by Sacks and Miller forcings. A connection with...

We prove that if ℱ is a non-meager P-filter, then both ℱ and ${}^{\omega}\mathcal{F}$ are countable dense homogeneous spaces.

We propose and study a “classification” of Borel ideals based on a natural infinite game involving a pair of ideals. The game induces a pre-order $\u2291$ and the corresponding equivalence relation. The pre-order is well founded and “almost linear”. We concentrate on ${F}_{\sigma}$ and ${F}_{\sigma \delta}$ ideals. In particular, we show that all ${F}_{\sigma}$-ideals are $\u2291$-equivalent and form the least equivalence class. There is also a least class of non-${F}_{\sigma}$ Borel ideals, and there are at least two distinct classes of ${F}_{\sigma \delta}$ non-${F}_{\sigma}$ ideals.

We answer a question of van Mill and Wattel by showing that there is a separable locally compact space which admits a continuous weak selection but is not weakly orderable. Furthermore, we show that a separable space which admits a continuous weak selection can be covered by two weakly orderable spaces. Finally, we give a partial answer to a question of Gutev and Nogura by showing that a separable space which admits a continuous weak selection admits a continuous selection for all finite sets.

We show that all sufficiently nice λ-sets are countable dense homogeneous (𝖢𝖣𝖧). From this fact we conclude that for every uncountable cardinal κ ≤ 𝔟 there is a countable dense homogeneous metric space of size κ. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size κ is equivalent to the existence of a λ-set of size κ. On the other hand, it is consistent with the continuum arbitrarily large that every 𝖢𝖣𝖧 metric space has size either ω₁ or 𝔠. An...

We prove there is a countable dense homogeneous subspace of ℝ of size ℵ₁. The proof involves an absoluteness argument using an extension of the ${L}_{\omega \u2081\omega}\left(Q\right)$ logic obtained by adding predicates for Borel sets.

We show that if $\mathcal{A}$ is an uncountable AD (almost disjoint) family of subsets of $\omega $ then the space $\Psi \left(\mathcal{A}\right)$ does not admit a continuous selection; moreover, if $\mathcal{A}$ is maximal then $\Psi \left(\mathcal{A}\right)$ does not even admit a continuous selection on pairs, answering thus questions of T. Nogura.

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