Cardinal invariants associated with the combinatorics of the uniformity number of the ideal of meager-additive sets

TL;DR

This work investigates the uniformity of the meager-additive ideal $\mathcal{MA}$ by introducing the new invariants $\mathfrak{b}^{\mathsf{eq}}_b$ and $\mathfrak{d}^{\mathsf{eq}}_b$ and embedding them into the Cichoń diagram via a relational-system framework and Tukey reductions. It proves key ZFC connections, notably $\mathrm{Cv}_{\mathcal{M}} \preceq_{\mathrm{T}} \mathsf{R}_b$ for $b \geq^* 2$, giving $\mathfrak{b}^{\mathsf{eq}}_b \leq \mathrm{non}(\mathcal{M})$ and $\mathrm{cov}(\mathcal{M}) \leq \mathfrak{d}^{\mathsf{eq}}_b$, and establishes $\mathrm{non}(\mathcal{MA}) = \min_b \mathfrak{b}^{\mathsf{eq}}_b$. The paper also develops forcing tools, notably $\mathbb{P}_b$, which are uniformly $\sigma$-uf-$\lim$-linked and increase $\mathfrak{b}^{\mathsf{eq}}_b$, to realize targeted configurations in Cichoń's diagram via FS iterations. These methods yield models with separations such as $\mathfrak{b}^{\mathsf{eq}}_b > \mathrm{cov}(\mathcal{N})$ or $\mathfrak{d}^{\mathsf{eq}}_b \leq \mathrm{non}(\mathcal{N})$, while Open Problems ask whether further strict inequalities among $\mathfrak{b}^{\mathsf{eq}}_b$, $\mathfrak{d}^{\mathsf{eq}}_b$, and classical invariants are consistent and how uf-limits might resolve deeper chains in the diagram.

Abstract

In [CMRM24], it was proved that it is relatively consistent that \emph{bounding number} $\mathfrak{b}$ is smaller than the uniformity of $\mathcal{MA}$, where $\mathcal{MA}$ denotes the ideal of the meager-additive sets of $2^ω$. To establish this result, a specific cardinal invariant, which we refer to as $\mathfrak{b}_b^\mathsf{eq}$, was introduced in close relation to Bartoszyński's and Judah's characterization of the uniformity of $\mathcal{MA}$. This survey aims to explore this cardinal invariant along with its dual, which we call as $\mathfrak{d}_b^\mathsf{eq}$. In particular, we will illustrate its connections with the cardinals represented in Cichoń's diagram. Furthermore, we will present several open problems pertaining to these cardinals.

Figures (5)

• Figure 1: Diagram of the cardinal invariants associated with $\mathcal{I}$. An arrow $\mathfrak x\rightarrow\mathfrak y$ means that (provably in ZFC) $\mathfrak x\le\mathfrak y$.
• Figure 2: Including $\mathfrak{b}^{\mathsf{eq}}_b$ and $\mathfrak{d}^{\mathsf{eq}}_b$ to Cichoń's diagram.
• Figure 3: Including $\mathfrak{b}^{\mathsf{eq}}_b$ and $\mathfrak{d}^{\mathsf{eq}}_b$ to Cichoń's diagram. Additionally, if $\sum_{k<\omega}\frac{1}{b(k)} = \infty$ then $\mathop{\mathrm{\hbox{\rm cov}}}\nolimits(\mathcal{N})\leq\mathfrak{b}_b^\mathsf{eq}$ and $\mathfrak{d}_b^\mathsf{eq}\leq\mathop{\mathrm{\hbox{\rm non}}}\nolimits(\mathcal{N})$.
• Figure 4: The constellation of Cichoń's diagram forced in \ref{['thm:a6']}.
• Figure 5: Cichoń's diagram after adding $\pi$-many generic reals with $\mathbb{P}_b$, where $\pi$ has uncountable cofinality and $|\pi|^{\aleph_0}=|\pi|$.

Theorems & Definitions (72)

• Definition 1.1
• Theorem 1.2: shmn
• Lemma 1.3: CMR2
• Definition 1.4
• Theorem 1.5: paw85, see also CMlocalc
• Lemma 1.6: CM
• Corollary 1.7
• Lemma 1.8: CMR2
• Theorem 1.9: paw85
• Theorem 1.10: bartJudah