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The additivity spectrum $ADD\left(ℐ\right)$ of an ideal $ℐ\subset 𝒫\left(I\right)$ is the set of all regular cardinals $\kappa$ such that there is an increasing chain $\{A_{\alpha }:\alpha <\kappa \}\subset ℐ$ with ${\bigcup }_{\alpha <\kappa }{A}_{\alpha }\notin ℐ$. We investigate which set $A$ of regular cardinals can be the additivity spectrum of certain ideals. Assume that $ℐ=ℬ$ or $ℐ=𝒩$, where $ℬ$ denotes the $\sigma$-ideal generated by the compact subsets of the Baire space ${\omega }^{\omega }$, and $𝒩$ is the ideal of the null sets. We show that if $A$ is a non-empty progressive set of uncountable regular cardinals and $pcf\left(A\right)=A$, then $ADD\left(ℐ\right)=A$ in some c.c.c generic extension of the...

We show that if a colouring c establishes ω₂ ↛ [(ω₁:ω)]² then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω₂]² → 2 establishing ω₂ ↛ [(ω₁:ω)]₂ such that some colouring g:[ω₁]² → 2 does not embed into c. It is also consistent that $2^{\omega ₁}$ is arbitrarily large, and there...

If there is no inner model with measurable cardinals, then for each cardinal $\lambda$ there is an almost disjoint family ${𝒜}_{\lambda }$ of countable subsets of $\lambda$ such that every subset of $\lambda$ with order type $\ge {\omega }^2$ contains an element of ${𝒜}_{\lambda }$.

A graph $G$ on ${\omega }_1$ is called $<\omega$- if for each uncountable $W\subset {\omega }_1$, $G$ is isomorphic to $G[W\setminus W^{\text{'}}]$ for some finite $W^{\text{'}}\subset W$. We show that in various models of ZFC if a graph $G$ is $<\omega$-smooth, then $G$ is necessarily trivial, i.eėither complete or empty. On the other hand, we prove that the existence of a non-trivial, $<\omega$-smooth graph is also consistent with ZFC.

Under $M{A}_{{\omega }_1}$ every uncountable almost disjoint family is either anti-Luzin or has an uncountable Luzin subfamily. This fails under CH. Related properties are also investigated.

We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang’s Conjecture for ${\aleph }_{\omega }$, we can find a counter-example to the characterization (Theorem 12). We then show that, in the model obtained by adding Cohen reals,...

Given an ideal $ℐ$ on $\omega$ let $𝔞\left(ℐ\right)$ ($\overline{𝔞}\left(ℐ\right)$) be minimum of the cardinalities of infinite (uncountable) maximal $ℐ$-almost disjoint subsets of ${\left[\omega \right]}^{\omega }$. We show that $𝔞\left({ℐ}_h\right)>\omega$ if ${ℐ}_h$ is a summable ideal; but $𝔞\left({𝒵}_{\overrightarrow{\mu }}\right)=\omega$ for any tall density ideal ${𝒵}_{\overrightarrow{\mu }}$ including the density zero ideal $𝒵$. On the other hand, you have $𝔟\le \overline{𝔞}\left(ℐ\right)$ for any analytic $P$-ideal $ℐ$, and $\overline{𝔞}\left({𝒵}_{\overrightarrow{\mu }}\right)\le 𝔞$ for each density ideal ${𝒵}_{\overrightarrow{\mu }}$. For each ideal $ℐ$ on $\omega$ denote ${𝔟}_{ℐ}$ and ${𝔡}_{ℐ}$ the unbounding and dominating numbers of $\langle {\omega }^{\omega },{\le }_{ℐ}\rangle$ where $f{\le }_{ℐ}g$ iff $\{n\in \omega :f(n)>g(n\left)\right\}\in ℐ$. We show that ${𝔟}_{ℐ}=𝔟$ and ${𝔡}_{ℐ}=𝔡$ for each analytic $P$-ideal $ℐ$. Given a Borel ideal $ℐ$ on...

Every crowded space $X$ is $\omega$-resolvable in the c.c.c. generic extension $V^{Fn\left(\right|X|,2)}$ of the ground model. We investigate what we can say about $\lambda$-resolvability in c.c.c. generic extensions for $\lambda >\omega$. A topological space is monotonically ${\omega }_1$-resolvable if there is a function $f:X\to {\omega }_1$ such that $$\{x\in X:f\left(x\right)\ge \alpha \}{\subset }^{dense}X$$ for each $\alpha <{\omega }_1$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is ${\omega }_1$-resolvable in some c.c.c. generic extension; (2) $X$ is monotonically ${\omega }_1$-resolvable; (3) $X$ is ${\omega }_1$-resolvable in the Cohen-generic extension $V^{Fn({\omega }_1,2)}$....

Let X be a set, κ be a cardinal number and let ℋ be a family of subsets of X which covers each x ∈ X at least κ-fold. What assumptions can ensure that ℋ can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover ℋ: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ℝⁿ by polyhedra and by arbitrary convex sets. We focus on...

We improve some results of Pavlov and Filatova, concerning a problem of Malykhin, by showing that every regular space X that satisfies Δ(X) > e(X) is ω-resolvable. Here Δ(X), the dispersion character of X, is the smallest size of a non-empty open set in X, and e(X), the extent of X, is the supremum of the sizes of all closed-and-discrete subsets of X. In particular, regular Lindelöf spaces of uncountable dispersion character are ω-resolvable. We also prove that any regular...

We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] by showing that ∙ there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable π-base (in fact, the minimum order of a π-base of the space can be made arbitrarily large); ∙ if there is a κ-Suslin line then there is a first countable GO-space of cardinality κ⁺ in which the order of any π-base is at least κ; ∙ it is consistent to have a first countable,...

Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put ${}_{\lambda }\left(\alpha \right)=s\in \left(\alpha \right):s\left(0\right)=\lambda =min\left[s\right(\beta ):\beta <\alpha ]$. We show that f ∈ (α) iff for some natural number n there are infinite cardinals $\lambda ₀i>\lambda ₁>...>{\lambda }_{n-1}$ and ordinals $\alpha ₀,...,{\alpha }_{n-1}$ such that $\alpha =\alpha ₀+\cdots +{\alpha }_{n-1}$ and $f=f₀⏜f₁⏜...⏜f_{n-1}$ where each $f_i{\in }_{{\lambda }_i}\left({\alpha }_i\right)$. Under GCH we prove that if α < ω₂ then (i) ${}_{\omega }\left(\alpha \right)=s{\in }^{\alpha }\omega ,\omega ₁:s\left(0\right)=\omega$; (ii) if λ > cf(λ) = ω, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega ₁-closedin\alpha$; (iii) if cf(λ) = ω₁, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega -closedandsuccessor-closedin\alpha$; (iv) if cf(λ) > ω₁, ${}_{\lambda }\left(\alpha \right){=}^{\alpha }\lambda$. This yields a complete characterization of the classes (α) for all α < ω₂,...

We investigate whether an arbitrary base for a dense-in-itself topological space can be partitioned into two bases. We prove that every base for a $T_3$ Lindelöf topology can be partitioned into two bases while there exists a consistent example of a first-countable, 0-dimensional, Hausdorff space of size $2^{\omega }$ and weight ${\omega }_1$ which admits a point countable base without a partition to two bases.

Let us denote by $\Phi (\lambda ,\mu )$ the statement that $𝔹\left(\lambda \right)=D{\left(\lambda \right)}^{\omega }$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $ℂ$ in $𝔹\left(\lambda \right)$ picks up all the $\mu$ colors. We call a space $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V}\subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and only if it...

We show that all finite powers of a Hausdorff space $X$ do not contain uncountable weakly separated subspaces iff there is a c.c.c poset $P$ such that in $V^P$ $X$ is a countable union of $0$-dimensional subspaces of countable weight. We also show that this theorem is sharp in two different senses: (i) we cannot get rid of using generic extensions, (ii) we have to consider all finite powers of $X$.

We show that if we add any number of Cohen reals to the ground model then, in the generic extension, a locally compact scattered space has at most ${\left(2^{\aleph ₀}\right)}^V$ levels of size ω. We also give a complete ZFC characterization of the cardinal sequences of regular scattered spaces. Although the classes of regular and of 0-dimensional scattered spaces are different, we prove that they have the same cardinal sequences.

Let $𝔾$ and $𝔻$, respectively, denote the partially ordered sets of homomorphism classes of finite undirected and directed graphs, respectively, both ordered by the homomorphism relation. Order theoretic properties of both have been studied extensively, and have interesting connections to familiar graph properties and parameters. In particular, the notion of a duality is closely related to the idea of splitting a maximal antichain. We construct both splitting and non-splitting infinite maximal antichains...