arXiv subjects

Diego A. Mejía

Publications and source records attributed to Diego A. Mejía.

24 records · Page 2Linked to original sources

arXiv2018

Matrix iterations with vertical support restrictions

We use coherent systems of FS iterations on a power set, which can be seen as matrix iteration that allows restriction on arbitrary subsets of the vertical component, to prove general theorems about preservation of certain type of unbounded families on definable structures and of certain mad families (like those added by Hechler's poset for adding an a.d. family) regardless of the cofinality of their size. In particular, we define a class of posets called $σ$-Frechet-linked and show that they work well to preserve mad families, and unbounded families on $ω^ω$. As applications of this method, we show that a large class of FS iterations can preserve the mad family added by Hechler's poset (regardless of the cofinality of its size), and the consistency of a constellation of Cichoń's diagram with 7 values where two of these values are singular.

math.LO↗

arXiv2018

On cardinal characteristics of Yorioka ideals

Yorioka [J. Symbolic Logic 67(4):1373-1384, 2002] 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 ZFC. We construct a matrix iteration of ccc posets to force that, for many ideals in that class, their associated cardinal invariants (i.e. additivity, covering, uniformity and cofinality) are pairwise different. In addition, we show that, consistently, the additivity and cofinality of Yorioka ideals does not coincide with the additivity and cofinality (respectively) of the ideal of Lebesgue measure zero subsets of the real line.

math.LO↗

arXiv2017

Coherent systems of finite support iterations

We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cichoń's diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a $Δ^1_3$ well-order of the reals.

math.LO↗

arXiv2016

Splitting, Bounding, and Almost Disjointness can be quite Different

We prove the consistency of $\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{p}=\mathfrak{g}=\mathfrak{s}<\mathrm{add}(\mathcal{M})=\mathrm{cof}(\mathcal{M})<\mathfrak{a}=\mathrm{non}(\mathcal{N})=\mathfrak{c}$ with ZFC where each of these cardinal invariants assume arbitrary uncountable regular values.

math.LO↗

arXiv2015

The left side of Cichoń's diagram

Using a finite support iteration of ccc forcings, we construct a model of $\aleph_1<\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{b}<\mathrm{non}(\mathcal{M})<\mathrm{cov}(\mathcal{M})=\mathfrak{c}$.

math.LO↗

arXiv2014

Rothberger gaps in fragmented ideals

The~\emph{Rothberger number} $\mathfrak{b} (\mathcal{I})$ of a definable ideal $\mathcal{I}$ on $ω$ is the least cardinal $κ$ such that there exists a Rothberger gap of type $(ω,κ)$ in the quotient algebra $\mathcal{P} (ω) / \mathcal{I}$. We investigate $\mathfrak{b} (\mathcal{I})$ for a subclass of the $F_σ$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is $\aleph_1$ while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum many) different Rothberger numbers associated with fragmented ideals.

math.LO↗