arXiv subjects

Vera Fischer

Publications and source records attributed to Vera Fischer.

At least 19 recordsLinked to original sources

arXiv2026

Strong Projective Witnesses

We show Shelah's original creature forcing from 1984 strongly preserves tight mad families. In particular, answering questions of Fischer and Friedman and Friedman and Zdomskyy, we show the constellation $\aleph_1 = \mathfrak{a} < \mathfrak{s} = \aleph_2$ is consistent with the existence of a $\Delta_3^1$ wellorder of the reals and tight mad families of sizes $\aleph_1, \aleph_2$ which are $\Pi_1^1, \Pi_2^1$-definable, respectively. Each of these projective definitions is of minimal possible complexity.

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arXiv2023

Cofinitary groups and projective well-orders

We introduce the notion of a tight cofinitary group, which captures forcing indestructibility of maximal cofinitary groups for a long list of partial orders, including Cohen, Sacks, Miller, Miller partition forcing and Shelah's poset for diagonalizing maximal ideal. Introducing a new robust coding technique, we establish the relative consistency of $\mathfrak{a}_g=\mathfrak{d}<\mathfrak{c}=\aleph_2$ alongside the existence of a $\Delta^1_3$-wellorder of the reals and a co-analytic witness for $\mathfrak{a}_g$.

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arXiv2023

Partitions of Baire space into compact sets

Under $\text{CH}$ we construct a partition of Baire space into compact sets, which is indestructible by countably supported iteration and product of Sacks forcing of any length, answering a question of Newelski. Further, we present an in-depth isomorphism-of-names argument for $\text{spec}(\mathfrak{a}_\text{T}) = \{\aleph_1, \mathfrak{c}\}$ in the product-Sacks model. Finally, we prove that Shelah's ultrapower model for the consistency of $\mathfrak{d} < \mathfrak{a}$ also satisfies $\mathfrak{a} = \mathfrak{a}_\text{T}$. Thus, consistently $\aleph_1 < \mathfrak{d} < \mathfrak{a} = \mathfrak{a}_\text{T}$ holds relative to a measurable.

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arXiv2023

Universally Sacks-indestructible combinatorial families of reals

We introduce the notion of an arithmetical type of combinatorial family of reals, which serves to generalize different types of families such as mad families, maximal cofinitary groups, ultrafilter bases, splitting families and other similar types of families commonly studied in combinatorial set theory. We then prove that every combinatorial family of reals of arithmetical type, which is indestructible by the product of Sacks forcing $\mathbb{S}^{\aleph_0}$, is in fact universally Sacks-indestructible, i.e. it is indestructible by any countably supported iteration or product of Sacks-forcing of any length. Further, under $\text{CH}$ we present a unified construction of universally Sacks-indestructible families for various arithmetical types of families. In particular we prove the existence of a universally Sacks-indestructible maximal cofinitary group under $\text{CH}$.

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arXiv2023

Realizing arbitrarily large spectra of $\mathfrak{a}_{\text{T}}$

We improve the state-of-the-art proof techniques for realizing various spectra of $\mathfrak{a}_{\text{T}}$ in order to realize arbitrarily large spectra. Thus, we make significant progress in addressing a question posed by Brian in his recent work. As a by-product, we obtain many complete subforcings and an algebraic analysis of the automorphisms of the forcing which adds a witness for the spectrum of $\mathfrak{a}_{\text{T}}$ of desired size.

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arXiv2023

Generic Selective Independent Families

We prove that the generic maximal independent family obtained by iteratively forcing with the Mathias forcing relative to diagonalization filters is densely maximal. Moreover, by choosing the filters with some care one can ensure the family is selective and hence forcing indestructible in a strong sense. Using this we prove that under $\mathfrak{p} = 2^{\aleph_0}$ there are selective independent families and also we show how to add selective independent families of any desired size.

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arXiv2023

Filters, ideal independence and ideal Mr\'owka spaces

A family $\mathcal{A} \subseteq [\omega]^\omega$ such that for all finite $\{X_i\}_{i\in n}\subseteq \mathcal A$ and $A \in \mathcal{A} \setminus \{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i \in n} X_i$ is infinite, is said to be ideal independent. We prove that an ideal independent family $\mathcal{A}$ is maximal if and only if $\mathcal A$ is $\mathcal J$-completely separable and maximal $\mathcal J$-almost disjoint for a particular ideal $\mathcal J$ on $\omega$. We show that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, where $\mathfrak{s}_{mm}$ is the minimal cardinality of maximal ideal independent family. This, in particular, establishes the independence of $\mathfrak{s}_{mm}$ and $\mathfrak{i}$. Given an arbitrary set $C$ of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality $\lambda$ for each $\lambda\in C$, thus establishing the consistency of $C\subseteq \hbox{spec}(\mathfrak{s}_{mm})$. Assuming $\mathsf{CH}$, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, $^\omega\omega$-bounding, $p$-point preserving forcing notion and evaluate $\mathfrak{s}_{mm}$ in several well studied forcing extensions. We also study natural filters associated with ideal independence and introduce an analog of Mr\'owka spaces for ideal independent families.

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arXiv2022

A co-analytic Cohen indestructible maximal cofinitary group

Assuming that every set is constructible, we find a $Π^1_1$ maximal cofinitary group of permutations of $\mathbb N$ which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans' result that there exists a $Π^1_1$ maximal cofinitary group in $L$.

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arXiv2022

Cohen Preservation and Independence

We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the independence number $\mathfrak{i}$ is strictly below $\mathfrak{c}$, including iterations of Sacks forcing, Miller partition forcing, $h$-perfect tree forcings, coding with perfect trees. Moreover, applying the theorem, we show that $\mathfrak{i} = \aleph_1$ in the Miller Lite model. An important aspect of the preservation theorem is the notion of "Cohen preservation", which we discuss in detail.

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arXiv2022

Filters and Ideal Independence

A family $\mathscr{I} \subseteq [ω]^ω$ such that for all finite $\{X_i\}_{i\in n}\subseteq \mathcal I$ and $A \in \mathscr{I} \setminus \{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i < n} X_i$ is infinite, is said to be ideal independent. An ideal independent family which is maximal under inclusion is said to be a maximal ideal independent family and the least cardinality of such family is denoted $\mathfrak{s}_{mm}$. We show that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, which in particular establishes the independence of $\mathfrak{s}_{mm}$ and $\mathfrak{i}$. Given an arbitrary set $C$ of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality $λ$ for each $λ\in C$, thus establishing the consistency of $C\subseteq \hbox{spec}(\mathfrak{s}_{mm})$. Assuming $\mathsf{CH}$, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, $^ωω$-bounding, $p$-point preserving forcing notion and evaluate $\mathfrak{s}_{mm}$ in several well studied forcing extensions.

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arXiv2022

Higher Independence

We study higher analogues of the classical independence number on $ω$. For $κ$ regular uncountable, we denote by $i(κ)$ the minimal size of a maximal $κ$-independent family. We establish ZFC relations between $i(κ)$ and the standard higher analogues of some of the classical cardinal characteristics, e.g. $\mathfrak{r}(κ)\leq\mathfrak{i}(κ)$ and $\mathfrak{d}(κ)\leq\mathfrak{i}(κ)$. For $κ$ measurable, assuming that $2^κ=κ^+$ we construct a maximal $κ$-independent family which remains maximal after the $κ$-support product of $λ$ many copies of $κ$-Sacks forcing. Thus, we show the consistency of $κ^+=\mathfrak{d}(κ)=\mathfrak{i}(κ)<2^κ$. We conclude the paper with interesting open questions and discuss difficulties regarding other natural approaches to higher independence.

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arXiv2022

Games on base matrices

Using a game characterization of distributivity, we show that base matrices for $\mathcal{P}(ω)/\text{fin}$ of regular height larger than $\mathfrak{h}$ necessarily have maximal branches which are not cofinal.

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arXiv2022

Projective well-orders and coanalytic witnesses

We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable $P$-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which $\mathfrak{a}=\mathfrak{u}=\mathfrak{i}=\aleph_1<2^{\aleph_0}=\aleph_2$, each of $\mathfrak{a}$, $\mathfrak{u}$, $\mathfrak{i}$ has a $Π^1_1$ witness and there is a $Δ^1_3$ well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a $Δ^1_3$ well-order of the reals is consistent with $\mathfrak{c}=\aleph_2$ and each of the following: $\mathfrak{a}=\mathfrak{u}<\mathfrak{i}$, $\mathfrak{a}=\mathfrak{i}<\mathfrak{u}$, $\mathfrak{a}<\mathfrak{u}=\mathfrak{i}$, where the smaller cardinal characteristics have co-analytic witnesses. Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type.

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arXiv2022

On heights of distributivity matrices

We construct a model in which there exists a distributivity matrix of regular height $λ$ larger than $\mathfrak{h}$; both $λ= \mathfrak{c}$ and $λ< \mathfrak{c}$ are possible. A distributivity matrix is a refining system of mad families without common refinement. Of particular interest in our proof is the preservation of $\mathcal{B}$-Canjarness.

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arXiv2021

Tight Eventually Different Families

Generalizing the notion of a tight almost disjoint family, we introduce the notions of a {\em tight eventually different} family of functions in Baire space and a {\em tight eventually different set of permutations} of $\omega$. Such sets strengthen maximality, exist under $\mathsf{MA} (\sigma {\rm -linked})$ and come with a properness preservation theorem. The notion of tightness also generalizes earlier work on the forcing indestructibility of maximality of families of functions. As a result we compute the cardinals $\mathfrak{a}_e$ and $\mathfrak{a}_p$ in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including $\mathfrak{a}_e = \mathfrak{a}_p = \mathfrak{d} < \mathfrak{a}_T$, $\mathfrak{a}_e = \mathfrak{a}_p < \mathfrak{d} = \mathfrak{a}_T$, $\mathfrak{a}_e = \mathfrak{a}_p = \mathfrak{u} < non(\mathcal N) = cof(\mathcal N)$ and $\mathfrak{a}_e = \mathfrak{a}_p =\mathfrak{i} < \mathfrak{u}$. We also show that there are $\Pi^1_1$ tight eventually different families and tight eventually different sets of permutations in $L$ thus obtaining the above inequalities alongside $\Pi^1_1$ witnesses for $\mathfrak{a}_e = \mathfrak{a}_p = \aleph_1$. Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic.

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arXiv2021

The Structure of $κ$-Maximal Cofinitary Groups

We study $κ$-maximal cofinitary groups for $κ$ regular uncountable, $κ= κ^{<κ}$. Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell's theorem, we show that: 1. Any $κ$-maximal cofinitary group has ${<}κ$ many orbits under the natural group action of $S(κ)$ on $κ$. 2. If $\mathfrak{p}(κ) = 2^κ$ then any partition of $κ$ into less than $κ$ many sets can be realized as the orbits of a $κ$-maximal cofinitary group. 3. For any regular $λ> κ$ it is consistent that there is a $κ$-maximal cofinitary group which is universal for groups of size ${<}2^κ= λ$. If we only require the group to be universal for groups of size $κ$ then this follows from $\mathfrak{p}(κ) = 2^κ$.

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arXiv2021

Strong independence and its spectrum

For $μ, κ$ infinite, say $\mathcal{A}\subseteq [κ]^κ$ is a $(μ,κ)$-maximal independent family if whenever $\mathcal{A}_0$ and $\mathcal{A}_1$ are pairwise disjoint non-empty in $[\mathcal{A}]^{<μ}$ then $\bigcap\mathcal{A}_0\backslash\bigcup\mathcal{A}_1 \not= \emptyset$, $\mathcal{A}$ is maximal under inclusion among families with this property, and moreover all such Booelan combinations have size $κ$. We denote by $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ the set of all cardinalities of such families, and if non-empty, we let $\mathfrak{i}_μ(κ)$ be its minimal element. Thus, $\mathfrak{i}_μ(κ)$ (if defined) is a natural higher analogue of the independence number on $ω$ for the higher Baire spaces. In this paper, we study $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ for $μ,κ$ uncountable. Among others, we show that: (1) The property $\mathfrak{sp}_{\mathfrak i}(μ,κ)\neq\emptyset$ cannot be decided on the basis of ZFC plus large cardinals. (2) Relative to a measurable, it is consistent that: (a) $(\exists κ{>}ω) \, \mathfrak{i}_κ(κ)<2^κ$; (b) $(\exists κ{>}ω)\,κ^+<\mathfrak{i}_{ω_1}(κ)<2^κ$. To the best knowledge of the authors, this is the first example of a $(μ,κ)$-maximal independent family of size strictly between $κ^+$ and $2^κ$, for uncountable $κ$. (3) $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ cannot be quite arbitrary.

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arXiv2020

Parallel non-linear iterations

Developing a system of parallel non-linear iterations, we establish the consistency of $\mathfrak{b}<\mathfrak{s}<\mathfrak{d}<\mathfrak{c}$ where $\mathfrak{b}, \mathfrak{d}, \mathfrak{c}$ are arbitrary subject to the known ZFC restrictions and $\mathfrak{s}$ is regular. By evaluating other invariants we achieve also the constellations $\mathfrak{b}<\mathfrak{r}<\mathfrak{d}<\mathfrak{c}$, $\mathfrak{b}<\mathfrak{e}<\mathfrak{d}<\mathfrak{c}$ and $\mathfrak{b}<\mathfrak{u}<\mathfrak{d}<\mathfrak{c}$.

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