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Miguel A. Cardona

Publications and source records attributed to Miguel A. Cardona.

At least 19 recordsLinked to original sources
arXiv2026

Revisiting $\mathfrak b$ and $\mathfrak d$ through Interval Structures

We investigate a family of relational systems arising from interval partitions of $ω$, inspired by Vojtáš's characterization of the bounding and dominating numbers. By varying the underlying asymptotic quantifiers and interval constraints, we obtain several natural interval-type generalizations. We show that the universal variants are remarkably robust: in all the discrete, colored, restricted, bounded, and measure-theoretic settings considered here, the associated bounding and dominating numbers coincide with the classical invariants $\mathfrak b$ and $\mathfrak d$. In contrast, the existential variants systematically reverse these invariants, yielding that the bounding number coincides with $\mathfrak d$ and the dominating number coincides with $\mathfrak b$.

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arXiv2026

Directed schemes of ideals and cardinal characteristics, I: the meager additive ideal

We introduce the notion of directed scheme of ideals to characterize peculiar ideals on the reals, which comes from a formalization of the framework of Yorioka ideals for strong measure zero sets. We prove general theorems for directed schemes and propose a directed scheme $\vec{\mathcal{M}} = \{\mathcal{M}_I \colon I\in\mathbb{I}\}$ for the ideal $\mathcal{MA}$ of meager-additive sets of reals. This directed scheme does not only helps us to understand more the combinatorics of $\mathcal{MA}$ and its cardinal characteristics, but provides us new characterizations of the additivity and cofinality numbers of the meager ideal of the reals. In addition, we display connections between the characteristics associated with $\mathcal{M}_I$ and other classical characteristics. Furthermore, we demonstrate the consistency of $\mathrm{cov}(\mathcal{NA})<\mathfrak{c}$ and $\mathrm{cof}(\mathcal{MA})<\mathrm{non}(\mathcal{SN})$. The first one answers a question raised by the authors in arXiv:2401.15364.

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arXiv2026

Slalom numbers

The paper is an extensive and systematic study of cardinal invariants we call slalom numbers, describing the combinatorics of sequences of sets of natural numbers. Our general approach, based on relational systems, covers many such cardinal characteristics, including localization and anti-localization cardinals. We show that most of the slalom numbers are connected to topological selection principles, in particular, we obtain the representation of the uniformity of meager and the cofinality of measure. Considering instances of slalom numbers parametrized by ideals on natural numbers, we focus on monotonicity properties with respect to ideal orderings and computational formulas for the disjoint sum of ideals. Hence, we get such formulas for several pseudo-intersection numbers as well as for the bounding and dominating numbers parametrized with ideals. Based on the effect of adding a Cohen real, we get many consistent constellations of different values of slalom numbers.

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arXiv2025

Finitely additive measures on Boolean algebras

In this article, we conduct a detailed study of \emph{finitely additive measures} (fams) in the context of Boolean algebras, focusing on three specific topics: freeness and approximation, existence and extension criteria, and integration theory. In the first topic, we present a classification of \emph{free} finitely additive measures, that is, those for which the measure of finite sets is zero, in terms of approximation to uniform probability measures. This inspires a weaker version of this notion, which we call the \emph{uniform approximation property}, characterized in terms of freeness and another well-determined type of fams we call \emph{uniformly supported}. In the second topic, we study new criteria for the existence and extension of finitely additive measures. In particular, we provide an extension of the so-called \emph{compatibility theorem} -- which characterizes when two finitely additive measures can be extended -- yielding a precise and compact characterization of when three finitely additive measures can be simultaneously extended, under the assumption that one of them is an ultrafilter. Finally, we study a Riemann-type integration theory on fields of sets with respect to finitely additive measures, allowing us to extend and generalize some classical concepts and results from real analysis, such as Riemann integration over rectangles in $\mathbb{R}^{n}$ and the Jordan measure. We also generalize the extension criteria for fams allowing desired values of integrals of a given set of functions. At the end, we explore the connection between integration in fields of sets and the Lebesgue integration in the Stone space of the corresponding field, where we establish a characterization of integrability in the sense of the Lebesgue-Vitali theorem, which follows as a consequence of our results.

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arXiv2025

Uniformity numbers of the null-additive and meager-additive ideals

Denote by $\mathcal{NA}$ and $\mathcal{MA}$ the ideals of null-additive and meager-additive subsets of~$2^ω$, respectively. We prove in ZFC that $\mathrm{add}(\mathcal{NA})=\mathrm{non}(\mathcal{NA})$ and introduce a new (Polish) relational system to reformulate Bartoszyński's and Judah's characterization of the uniformity of $\mathcal{MA}$, which is helpful to understand the combinatorics of $\mathcal{MA}$ and to prove consistency results. As for the latter, we prove that $\mathrm{cov}(\mathcal{MA})<\mathfrak{c}$ (even $\mathrm{cov}(\mathcal{MA})<\mathrm{non}(\mathcal{N})$) is consistent with ZFC, as well as several constellations of Cichoń's diagram with $\mathrm{non}(\mathcal{NA})$, $\mathrm{non}(\mathcal{MA})$ and $\mathrm{add}(\mathcal{SN})$, which include $\mathrm{non}(\mathcal{NA})<\mathfrak{b}< \mathrm{non}(\mathcal{MA})$ and $\mathfrak{b}< \mathrm{add}(\mathcal{SN})<\mathrm{cov}(\mathcal{M})<\mathfrak{d}=\mathfrak{c}$.

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arXiv2025

Separating cardinal characteristics of the strong measure zero ideal

Let $\mathcal{SN}$ be the $σ$-ideal of the strong measure zero sets of reals. We present general properties of forcing notions that allow to control of the additivity of $\mathcal{SN}$ after finite support iterations. This is applied to force that the four cardinal characteristics associated with $\mathcal{SN}$ are pairwise different: \[\mathrm{add}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{non}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN}).\] Furthermore, we construct a forcing extension satisfying the above and Cichoń's maximum (i.e.\ that the non-dependent values in Cichoń's diagram are pairwise different).

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arXiv2025

Constant prediction and evasion number, I: Generalization and variants

Using the concept of constant evasion to different sorts of suitable binary relations, we establish many cardinal invariants derived from the established cardinal invariants $\mathfrak{e}^\mathrm{const}_{n}$ and $\mathfrak{v}^\mathrm{const}_{n}$, called the constant evasion number and the constant prediction number. We formulate several limits and consistency results pertaining to them.

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arXiv2025

Adding the constant evasion and constant prediction numbers to Cichoń's maximum

Let $\mathfrak{e}^\mathsf{const}_2$ be the constant evasion number, that is, the size of the least family $F\subseteq{}^ω2$ of reals such that for each predictor $π\colon {}^{<ω}2\to 2$ there is $x\in F$ which is not constantly predicted by $π$; and let $\mathfrak{v}_2^\mathsf{const}$ be the constant prediction number, that is, the size of the least family $Π_2$ of functions $π\colon {}^{<ω}2\to 2$ such that for each $x\in{}^ω2$ there is $π\inΠ_2$ that predicts constantly $x$. In this work, we show that the constant evasion number $\mathfrak{e}_2^{\mathrm{cons}}$ and the constant prediction number $\mathfrak{v}_2^\mathsf{const}$ can be added to Cichoń's maximum with distinct values.

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arXiv2025

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

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.

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arXiv2025

Cardinal characteristics associated with small subsets of reals

Inspired by Bartoszyński's work on small sets, we introduce a new ideal defined by interval partitions on natural numbers and summable sequences of positive reals. Similarly, we present another ideal that relies on Bartoszyński's and Shelah's representation of $F_σ$ measure zero sets. We show they are $σ$-ideals characterizing all small sets and $F_σ$ measure zero sets. We also study the cardinal characteristics associated with the introduced ideals. We use them to describe the invariants of measure, discuss their connection to Cichoń's diagram, and present related consistency results.

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arXiv2025

More separations of cardinal characteristics of the strong measure zero ideal

Let $\mathcal{N}$ be the $σ$-ideal of the null sets of reals. We introduce a new property of forcing notions that enable control of the additivity of $\mathcal{N}$ after finite support iterations. This is applied to answer some open questions from the work of Brendle, the first author, and Mejía~\cite{BCM2}.

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arXiv2024

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

We improve the previous work of Yorioka and the first author about the combinatorics of the ideal $\mathcal{SN}$ of strong measure zero sets of reals. We refine the notions of dominating systems of the first author and introduce the new combinatorial principle $\mathrm{DS}(δ)$ that helps to find simple conditions to deduce $\mathfrak{d}_κ\leq \mathrm{cof}(\mathcal{SN})$ (where $\mathfrak{d}_κ$ is the dominating number on $κ^κ$). In addition, we find a new upper bound of $\mathrm{cof}(\mathcal{SN})$ by using products of relational systems and cardinal characteristics associated with Yorioka ideals. In addition, we dissect and generalize results from Pawlikowski to force upper bounds of the covering of $\mathcal{SN}$, particularly for finite support iterations of precaliber posets. Finally, as applications of our main theorems, we prove consistency results about the cardinal characteristics associated with $\mathcal{SN}$ and the principle $\mathrm{DS}(δ)$. For example, we show that $\mathrm{cov}(\mathcal{SN})<\mathrm{non}(\mathcal{SN})=\mathfrak{c}<\mathrm{cof}(\mathcal{SN})$ holds in Cohen model, and we refine a result (and the proof) of the first author about the consistency of $\mathrm{cov}(\mathcal{SN})<\mathrm{non}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$, with $\mathfrak{c}$ in any desired position with respect to $\mathrm{cof}(\mathcal{SN})$, and the improvement that $\mathrm{non}(\mathcal{SN})$ can be singular here.

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arXiv2024

A general theory of iterated forcing using finitely additive measures

Based on the work of Shelah, Kellner, and Tănasie (Fund. Math., 166(1-2):109-136, 2000 and Comment. Math. Univ. Carolin., 60(1):61-95, 2019), and the recent developments in the third author's master's thesis, we develop a general theory of iterated forcing using finitely additive measures. For this purpose, we introduce two new notions: on the one hand, we define a new linkedness property, called $μ$-$\mathrm{FAM}$-linked and, on the other hand, we generalize the notion of intersection number to forcing notions, which justifies the limit steps of our iteration theory. Our theory also generalizes iterations with ultrafilters, which have played an important role in the proof of the consistency of Cichoń's maximum. We further show that any iteration constructed with our theory preserves strong unbounded families and what we call anti-Bendixson families, which play a central role in preserving witnesses of $\mathrm{cov}(\mathcal{N})$ of singular size (even of countable cofinality). We also show that our iteration method does not increase $\mathrm{non}(\mathcal{E})$, the smallest size of a set of reals that cannot be covered by an $F_σ$ measure zero set. Finally, we apply our theory to prove a new separation of the left-hand side of Cichoń's diagram where $\mathrm{cov}(\mathcal{N})$ is possibly singular, even with countable cofinality.

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arXiv2024

Soft-linkedness

We have revised the softness property introduced by Jörg Brendle and Haim Judah (perfect sets of random reals. Israel J. Math., 83(1-2):153-176, 1993), to present a new definition of a class of posets called $σ$-soft-linked. Our work demonstrates that these posets work well to preserve the evasion number as well as the bounding number small in generic extensions. Furthermore, we establish a connection between our concept and the Fréchet-linked notion introduced by Diego A. Mejía (Matrix iterations with vertical support restrictions. In Proceedings of the 14th1 and 15th Asian Logic Conferences, pages 213-248. World Sci. Publ., Hackensack, NJ, 2019).

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arXiv2023

Localization and anti-localization cardinals

This paper is intended to survey the basics of localization and anti-localization cardinals on the reals, and its interplay with notions and cardinal characteristics related to measure and category.

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arXiv2022

Forcing constellations of Cichoń's diagram by using the Tukey order

We use known finite support iteration techniques to present various examples of models where several cardinal characteristics of Cichoń's diagram are pairwise different. We show some simple examples forcing the left-hand side of Cichoń's diagram, and present the technique of restriction to models to force Cichoń's maximum (original from Goldstern, Kellner, Shelah, and the second author). We focus on how the values forced in all the constellations are obtained via the Tukey order.

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