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Vera Fischer

Publications and source records attributed to Vera Fischer.

34 records · Page 2Linked to original sources
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.

math.LO↗
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 $\mathfrak{b}<\mathfrak{u}<\mathfrak{d}<\mathfrak{c}$.

math.LO↗
arXiv2020

Non-linear iterations and higher splitting

We show that generalized eventually narrow sequences on a strongly inaccessible cardinal $κ$ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechler forcing on $κ$. Moreover assuming GCH, $κ^{<κ}=κ$, we show that: (1) if $κ$ is strongly unfoldable, $κ^+\leqβ=\hbox{cf}(β)\leq \hbox{cf}(δ)\leqδ\leqμ$ and $\hbox{cf}(μ)>κ$,then there is a cardinal preserving generic extension in which $$\mathfrak{s}(κ)=κ^+\leq\mathfrak{b}(κ)=β\leq\mathfrak{d}(κ)=δ\leq 2^κ=μ.$$ (2) if $κ$ is strongly inaccessible, $λ>κ^+$, then in the generic extension obtained as the $<κ$-support iteration of $κ$-Hechler forcing of length $λ$ there are no $κ$-towers of length $λ$.

math.LO↗
arXiv2019

Selective independence

Let $\mathfrak{i}$ denote the minimal cardinality of a maximal independent family and let $\mathfrak{a}_T$ denote the minimal cardinality of a maximal family of pairwise almost disjoint subtrees of $2^{<ω}$. Using a countable support iteration of proper, $^ωω$-bounding posets of length $ω_2$ over a model of CH, we show that consistently $\mathfrak{i}<\mathfrak{a}_T$. Moreover, we show that the inequality can be witnessed by a co-analytic maximal independent family of size $\aleph_1$ in the presence of a $Δ^1_3$ definable well-order of the reals. The main result of the paper can be viewed as a partial answer towards the well-known open problem of the consistency of $\mathfrak{i}<\mathfrak{a}$.

math.LO↗
arXiv2019

Towers and gaps at uncountable cardinals

Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either $\mathfrak p(κ)=\mathfrak t(κ)$ or there is a $(\mathfrak p(κ),λ)$-gap of club-supported slaloms for some $λ< \mathfrak p(κ)$. While the existence of such gaps is unclear, this is a promising step to lift Malliaris and Shelah's proof of $\mathfrak p=\mathfrak t$ to uncountable cardinals. We do analyze gaps of slaloms and, in particular, show that $\mathfrak p(κ)$ is always regular; the latter extends results of Garti. Finally, we turn to club variants of $\mathfrak p(κ)$ and present a new model for the inequality $\mathfrak{p}(κ) = κ^+ < \mathfrak{p}_{cl}(κ) = 2^κ$. In contrast to earlier arguments by Shelah and Spasojevic, we achieve this by adding $κ$-Cohen reals and then successively diagonalising the club-filter; the latter is shown to preserve a Cohen witness to $\mathfrak{p}(κ) = κ^+$.

math.LO↗
arXiv2019

Definable Maximal Independent Families

We study maximal independent families (m.i.f.) in the projective hierarchy. We show that (a) the existence of a $\boldsymbolΣ^1_2$ m.i.f. is equivalent to the existence of a $\boldsymbolΠ^1_1$ m.i.f., (b) in the Cohen model, there are no projective maximal independent families, and (c) in the Sacks model, there is a $\boldsymbolΠ^1_1$ m.i.f. We also consider a new cardinal invariant related to the question of destroying or preserving maximal independent families.

math.LO↗
arXiv2019

Free sequences in P(ω)/fin

We investigate maximal free sequences in the Boolean algebra $\mathcal{P}(ω)/\mathrm{fin}$, as defined by D. Monk. We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted $\mathfrak{f}$. Answering a question of Monk, we demonstrate the consistency of $ω_1 = \mathfrak{i} = \mathfrak{f} < \mathfrak{u} = ω_2$. In fact, this consistency is demonstrated in the model of S. Shelah for $\mathfrak{i} < \mathfrak{u}$. Our paper provides a streamlined and mostly self contained presentation of this construction.

math.LO↗
arXiv2018

More ZFC inequalities between cardinal invariants

Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show that if $κ=λ^+$ for some $λ\geq ω$ and $\mathfrak b(κ)=κ^+$ then $\mathfrak a_e(κ)=\mathfrak a_p(κ)=κ^+$. If, additionally, $2^{<λ}=λ$ then $\mathfrak a_g(κ)=κ^+$ as well. Furthermore, we prove a variety of new bounds for $\mathfrak d(κ)$ in terms of $\mathfrak r(κ)$, including $\mathfrak d(κ)\leq \mathfrak r_σ(κ)\leq \mathrm{cof}([\mathfrak r(κ)]^ω)$, and $\mathfrak d(κ)\leq \mathfrak r(κ)$ whenever $\mathfrak r(κ)<\mathfrak b(κ)^{+κ}$ or $\mathrm{cof}(\mathfrak r(κ))\leq κ$ holds.

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↗
arXiv2016

Definable maximal cofinitary groups

Using countable support iteration of $S$-proper posets, for some appropriate stationary set $S$, we obtain a generic extension of the constructible universe, in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and there is a maximal cofinitary group with a $Π^1_2$-definable set of generators.

math.LO↗
arXiv2013

Template iterations and maximal cofinitary groups

The main result of the present paper is that $\mathfrak a_g$, the minimal size of maximal cofinitary group, can be of countable cofinality. To prove this we define a natural poset for adding a maximal cofinitary group of a given cardinality, which enjoys certain combinatorial properties allowing it to be used within a similar template forcing construction. Additionally we obtain that $\mathfrak a_p$, the minimal size of a maximal family of almost disjoint permutations, and $\mathfrak a_e$, the minimal size of a maximal eventually different family, can be of countable cofinality.

math.LO↗
arXiv2011

Projective maximal families of orthogonal measures with large continuum

We study maximal orthogonal families of Borel probability measures on $2^ω$ (abbreviated m.o. families) and show that there are generic extensions of the constructible universe $L$ in which each of the following holds: (1) There is a $Δ^1_3$-definable well order of the reals, there is a $Π^1_2$-definable m.o. family, there are no $\mathbfΣ^1_2$-definable m.o. families and $\mathfrak{b}=\mathfrak{c}=ω_3$ (in fact any reasonable value of $\mathfrak{c}$ will do). (2) There is a $Δ^1_3$-definable well order of the reals, there is a $Π^1_2$-definable m.o. family, there are no $\mathbfΣ^1_2$-definable m.o. families, $\mathfrak{b}=ω_1$ and $\mathfrak{c}=ω_2$.

math.LO↗
arXiv2009

A co-analytic maximal set of orthogonal measures

We prove that if $V=L$ then there is a $Π^1_1$ maximal orthogonal (i.e. mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known Theorem of Preiss and Rataj that no analytic set of measures can be maximal orthogonal.

math.LO↗