The covering numbers of Mycielski ideals are all equal

Covering of the null ideal may have countable cofinality

We prove that it is consistent that the covering of the ideal of measure zero sets has countable cofinality.

The covering number of the strong measure zero ideal can be above almost everything else

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal $\mathcal{SN}$. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that $\mathrm{non}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$, which is the first consistency result where more than two cardinal invariants associated with $\mathcal…

On cofinality of the smallest covering of the real line by meager sets II

We study the ideal of meager sets and related ideals.

A covering theorem and the random-indestructibility of the density zero ideal

The main goal of this note is to prove the following theorem. If $A_n$ is a sequence of measurable sets in a $σ$-finite measure space $(X, \mathcal{A}, μ)$ that covers $μ$-a.e. $x \in X$ infinitely many times, then there exists a sequence of integers $n_i$ of density zero so that $A_{n_i}$ still covers $μ$-a.e. $x \in X$ infinitely many times. The proof is a probabilistic construction. As an application we give a simple direct proof of the known theorem that the ideal of density zero subsets of the natural numbers is random-indestructible, that is, random forcing does not add a co-infinite set…

关于强测度零集理想的共尾度与覆盖的进一步研究

More about the cofinality and the covering of the ideal of strong measure zero sets

本文通过新组合原理 $\mathrm{DS}(δ)$、关系系统乘积及力迫迭代，改进强测度零集理想 $\mathcal{SN}$ 的共尾度与覆盖数界，并证明多种基数特征分离的一致性。

Two inequalities between cardinal invariants

We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $ω$ of asymptotic density $0$. We obtain an upper bound on the $\ast$-covering number, sometimes also called the weak covering number, of this ideal by proving in Section \ref{sec:covz0} that ${\mathord{\mathrm{cov}}}^{\ast}({\mathcal{Z}}_{0}) \leq \mathfrak{d}$. In Section \ref{sec:skbk} we investigate the relationship between the bounding and splitting numbers at regular uncountable cardinals. We pro…