# Miguel Antonio CardonaPavol Jozef Šafárik University in Košice · Institute of Mathematics

Miguel Antonio Cardona

PhD

## About

17

Publications

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40

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Introduction

**Skills and Expertise**

## Publications

Publications (17)

Let $\mathcal{E}$ be the $\sigma$-ideal generated by the closed measure zero sets of reals. We use an ultrafilter-extendable matrix iteration of ccc posets to force that, for $\mathcal{E}$, their associated cardinal characteristics (i.e.\ additivity, covering, uniformity and cofinality) are pairwise different.

We use known finite support iteration techniques to present various examples of models where several cardinal characteristics of Cicho\'n's diagram are pairwise different. We show some simple examples forcing the left-hand side of Cicho\'n's diagram, and present the technique of restriction to models to force Cicho\'n's maximum (original from Golds...

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal 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....

Combining creature forcing approaches from arXiv:1003.3425 and arXiv:1402.0367, we show that, under CH, there is a proper $\omega^\omega$-bounding poset with $\aleph_2$-cc that forces continuum many pairwise different cardinal characteristics, parametrised by reals, for each one of the following six types: uniformity and covering numbers of Yorioka...

In J. Symbolic Logic,51(4): 957-968, 1986, Pawlikowski proved that, if $r$ is a random real over $\mathbf{N}$, and $c$ is Cohen real over $\mathbf{N}[r]$, then (a) in $\mathbf{N}[r][c]$ there is a Cohen real over $\mathbf{N}[c]$, and (b) $2^\omega\cap\mathbf{N}[c]\notin\mathcal{N}\cap\mathbf{N}[r][c]$, so in $\mathbf{N}[r][c]$ there is no random re...

In [Paw86] Pawlikowski proved that, if r is a random real over N, and c is Cohen real over N[r], then
(a) in N[r][c] there is a Cohen real over N[c], and
(b) 2^ω ∩ N[c] ∈ N∩N[r][c], so in N[r][c] there is no random real over N[c].
To prove this, Pawlikowski proposes the following notion: Given two models N ⊆ M of ZFC, we associate with a cardina...

We introduce the property “F-linked” of subsets of posets for a given free filter F on the natural numbers, and define the properties “μ-F-linked” and “θ-F-Knaster” for posets in a natural way. We show that θ-F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families.
Concerning iterations of such posets, w...

We show that the set of the ground-model reals has strong measure zero (is strongly meager) after adding a single Cohen real (random real). As consequence we prove that the set of the ground-model reals has strong measure zero after adding a single Hechler real.

Let SN be the strong measure zero σ-ideal. We prove a result providing bounds for cof(SN) which implies Yorioka's characterization of the cofinality of the strong measure zero. In addition, we use forcing matrix iterations to construct a model of ZFC that satisfies add(SN) = cov(SN) < non(SN) < cof(SN).

In this paper we present a simpler proof of the fact that no inequality between $\mathrm{cof}(\mathcal{SN})$ and $\mathfrak{c}$ can be decided in ZFC by using well-known tecniques and results.

In this paper we present a simpler proof of the fact that no inequality between cof(SN) and c can be decided in ZFC by using well-known tecniques and results.

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...

We introduce the property "$F$-linked" of subsets of posets for a given free filter $F$ on the natural numbers, and define the properties "$\mu$-$F$-linked" and "$\theta$-$F$-Knaster" for posets in the natural way. We show that $\theta$-$F$-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concernin...

In this poster we expose some recent results and open problems about the cardinals invariant cardinals associated with the Yorioka ideals.

Yorioka introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on the real line cannot be decided in . We construct a matrix iteration of c.c.c. posets to force that, for many ideals in that class, their associated card...