Jörg Brendle (2003) used Hechler’s forcing notion for adding a maximal almost disjoint family along an appropriate template forcing construction to show that (the minimal size of a maximal almost disjoint family) can be of countable cofinality. The main result of the present paper is that ${}_g$, the minimal size of a maximal cofinitary group, can be of countable cofinality. To prove this we define a natural poset for adding a maximal cofinitary group of a given cardinality, which enjoys certain combinatorial...

Using finite support iteration of ccc partial orders we provide a model of 𝔟 = κ < 𝔰 = κ⁺ for κ an arbitrary regular, uncountable cardinal.

We show that cov(M) is the least infinite cardinal λ such that $P_{\omega }\left(\lambda \right)$ (the set of all finite subsets of λ ) fails to satisfy a certain natural generalization of Ramsey’s Theorem.

The linear refinement number is the minimal cardinality of a centered family in ${\left[\omega \right]}^{\omega }$ such that no linearly ordered set in $({\left[\omega \right]}^{\omega },\subseteq *)$ refines this family. The linear excluded middle number is a variation of . We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classical combinatorial cardinal characteristics of the continuum. We prove that = = in all models where the continuum is at most ℵ₂, and that the cofinality of is...

We examine the splitting number (B) and the reaping number (B) of quotient Boolean algebras B = (ω)/ℐ where ℐ is an $F_{\sigma }$ ideal or an analytic P-ideal. For instance we prove that under Martin’s Axiom ((ω)/ℐ) = for all $F_{\sigma }$ ideals ℐ and for all analytic P-ideals ℐ with the BW property (and one cannot drop the BW assumption). On the other hand under Martin’s Axiom ((ω)/ℐ) = for all $F_{\sigma }$ ideals and all analytic P-ideals ℐ (in this case we do not need the BW property). We also provide applications of these characteristics...

We show the consistency of the statement: "the set of regular cardinals which are the characters of ultrafilters on ω is not convex". We also deal with the set of π-characters of ultrafilters on ω.

Let χ be the minimum cardinality of a subset of ${}^{\omega }2$ that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of a creature forcing we show that < χ is consistent. We thus answer a question by Vojtáš. We give two kinds of models for the strict inequality. The first is the combination of an ℵ₂-iteration of some proper forcing with adding ℵ₁ random reals. The second kind of models is obtained by adding δ random reals to a model of $M{A}_{<\kappa }$ for some δ ∈ [ℵ₁,κ). It...

The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\left\{b_n\right\}$ of positive elements of $B$, there is a sequence $\left\{{\lambda }_n\right\}$ of positive reals, and $b\in B$, with ${\lambda }_n{b}_n\le b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “$\sigma$” obtains for a Riesz space of continuous real-valued functions $C\left(X\right)$. A basic result is: For discrete $X$, $C\left(X\right)$ has $\sigma$ iff the cardinal $\left|X\right|<𝔟$, Rothberger’s bounding number. Consequences and...

In [Two examples of Borel partially ordered sets with the countable chain condition, Proc. Amer. Math. Soc. 112 (1991), no. 4, 1125–1128], Todorcevic introduced a ccc forcing which is Borel definable in a separable metric space. In [On Todorcevic orderings, Fund. Math., to appear], Balcar, Pazák and Thümmel applied it to more general topological spaces and called such forcings Todorcevic orderings. There they analyze Todorcevic orderings quite deeply. A significant remark is that Thümmel solved...

We study and classify topologically invariant σ-ideals with an analytic base on Euclidean spaces, and evaluate the cardinal characteristics of such ideals.

We formulate variants of the cardinals f, p and t in terms of families of measurable functions, in order to examine the effect upon these cardinals of adding one random real.

We compute transitive cardinal coefficients of ideals on generalized Cantor spaces. In particular, we show that there exists a null set $A\subseteq 2^{\omega ₁}$ such that for every null set $B\subseteq 2^{\omega ₁}$ we can find $x\in 2^{\omega ₁}$ such that A ∪ (A+x) cannot be covered by any translation of B.