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Heike Mildenberger

Publications and source records attributed to Heike Mildenberger.

At least 19 recordsLinked to original sources
arXiv2026

Good Scales and Non-Compactness of Squares

Cummings, Foreman, and Magidor investigated the extent to which square principles are compact at singular cardinals. The first author proved that if $κ$ is a singular strong limit of uncountable cofinality, all scales on $κ$ are good, and $\square^*_δ$ holds for all $δ<κ$, then $\square_κ^*$ holds. In this paper we will present a strongly contrasting result for $\aleph_ω$. We construct a model in which $\square_{\aleph_n}$ holds for all $n<ω$, all scales on $\aleph_ω$ are good, but in which $\square_{\aleph_ω}^*$ fails and some weak forms of internal approachability for $[H(\aleph_{ω+1})]^{\aleph_1}$ fail. This requires an extensive analysis of the dominating and approximation properties of a version of Namba forcing. We also prove some supporting results.

math.LO↗
arXiv2025

Forcing Diamond and Applications to Iterability

We show that higher Sacks forcing at a regular limit cardinal and club Miller forcing at an uncountable regular cardinal both add a diamond sequence. We answer the longstanding question, whether $κ= κ^{<κ} \geq\aleph_1$ implies that $κ$-supported iterations of $κ$-Sacks forcing do not collapse $κ^+$ and are $κ$-proper in the affirmative. The results pertain to other higher tree forcings.

math.LO↗
arXiv2021

Distributivity and Minimality in Perfect Tree Forcings for Singular Cardinals

Dobrinen, Hathaway and Prikry studied a forcing $\mathbb{P}_κ$ consisting of perfect trees of height $λ$ and width $κ$ where $κ$ is a singular $ω$-strong limit of cofinality $λ$. They showed that if $κ$ is singular of countable cofinality, then $\mathbb{P}_κ$ is minimal for $ω$-sequences assuming that $κ$ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption. Prikry proved that $\mathbb{P}_κ$ is $(ω,ν)$-distributive for all $ν<κ$ given a singular $ω$-strong limit cardinal $κ$ of countable cofinality, and Dobrinen et al$.$ asked whether this result generalizes if $κ$ has uncountable cofinality. We answer their question in the negative by showing that $\mathbb{P}_κ$ is not $(λ,2)$-distributive if $κ$ is a $λ$-strong limit of uncountable cofinality $λ$ and we obtain the same result for a range of similar forcings, including one that Dobrinen et al$.$ consider that consists of pre-perfect trees. We also show that $\mathbb{P}_κ$ in particular is not $(ω,\cdot,λ^+)$-distributive under these assumptions. While developing these ideas, we address natural questions regarding minimality and collapses of cardinals.

math.LO↗
arXiv2021

Mathias and Silver forcing parametrized by density

We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse 2^ωto ω, while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property.

math.LO↗
arXiv2020

On splitting trees

We investigate two variants of splitting tree forcing, their ideals and regularity properties. We prove connections with other well-known notions, such as Lebesgue measurablility, Baire- and Doughnut-property and the Marczewski field. Moreover, we prove that any \emph{absolute} amoeba forcing for splitting trees necessarily adds a dominating real, providing more support to Spinas' and Hein's conjecture that $\add(\ideal{I}_\spl) \leq \mathfrak{b}$.

math.LO↗
arXiv2019

Local Ramsey Spaces in Matet Forcing Extensions and Finitely Many Near-Coherence Classes

We introduce Gowers--Matet forcing with a finite sequence of pairwise non-isomorphic Ramsey ultrafilters over $ω$, and with this forcing we settle the long-standing problem of the spectrum of numbers near-coherence classes. We prove that for any finite $n \geq 1$, there is a forcing extension with exactly $n$ near-coherence classes of ultrafilters. For evaluating the new forcing, we prove a strengthening of Gowers's theorem on colourings of ${\rm Fin}_k$.

math.LO↗
arXiv2019

Uncountable structures are not classifiable up to bi-embeddability

Answering some of the main questions from [MR13], we show that whenever $κ$ is a cardinal satisfying $κ^{< κ} = κ> ω$, then the embeddability relation between $κ$-sized structures is strongly invariantly universal, and hence complete for ($κ$-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [LR05,FMR11,Wil14,CMR17].

math.LO↗
arXiv2019

A Version of $κ$-Miller Forcing

Let $κ$ be an uncountable cardinal such that $2^{<κ} = κ$ or just ${\rm cf}(κ) > ω$, $2^{2^{<κ}}= 2^κ$, and $([κ]^κ, \supseteq)$ collapses $2^κ$ to $ω$. We show under these assumptions the $κ$-Miller forcing with club many splitting nodes collapses $2^κ$ to $ω$ and adds a $κ$-Cohen real.

math.LO↗
arXiv2015

Cardinal invariants distinguishing permutation groups

We prove that for infinite cardinals $κ<λ$ the alternating group $Alt(λ)$ (of even permutations) of $λ$ is not embeddable into the symmetric group $Sym(κ)$ (of all permutations) of $κ$. To prove this fact we introduce and study several monotone cardinal group invariants which take value $κ$ on the groups $Alt(κ)$ and $Sym(κ)$.

math.GR↗
arXiv2015

The Filter Dichotomy Principle Does not Imply the Semifilter Trichotomy Principle

We answer Blass' question from 1989 of whether the inequality $\gu < \gro$ is strictly stronger than the filter dichotomy principle affirmatively. We show that there is a forcing extension in which every non-meagre filter on $ω$ is ultra by finite-to-one and the semifilter trichotomy does not hold. This trichotomy says: every semifilter is either meagre or comeagre or ultra by finite-to-one. The trichotomy is equivalent to the inequality $\gu<\gro$ by work of Blass and Laflamme. Combinatorics of block sequences is used to establish forcing notions that preserve suitable properties of block sequences.

math.LO↗
arXiv2015

A solution to Roitman's problem

We answer Question~3.2 from Shelah \cite{Sh:666}: Given a maximal almost disjoint (mad) family $\mathcal A$ of size $\aleph_1$, we construct a forcing ${\mathbb Q}(\mathcal A)$ that has Axiom A, is ${}^ωω$-bounding, preserves selective ultrafilters, has the $\aleph_2$-properness isomorphism condition (p.i.c.), and destroys the mad family $\mathcal A$. We develop a new construction technique for partial orders, combining ladder systems for $ω_1$ with trees of normed creatures. Countable support iteration of the new kind of iterands solves Roitman's problem in the case of $d=\aleph_1$ and also simultaneously the open question about the relative consistency of $u = \aleph_1 < a$: It is consistent relative to ZFC that there is a dominating set of size $\aleph_1$ and a selective ultrafilter with character $\aleph_1$ and the minimal size of a mad family is $\aleph_2$, like the continuum.

math.LO↗
arXiv2013

Many countable support iterations of proper forcings preserve Souslin trees

We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support interations in the so-called Case A that does not need a division into forcings that add reals and those who do not.

math.LO↗
arXiv2012

Splitting families and complete separability

We answer a question from Raghavan and Stepr{ā}ns' paper on weakly tight families by showing that $\mathfrak{s} = {\mathfrak{s}}_{ω, ω}$. Then we use this to construct a completely separable maximal almost disjoint family under $\s \leq \a$, partially answering a question of Shelah.

math.LO↗
arXiv2010

Covering the Baire space by families which are not finitely dominating

It is consistent (relative to ZFC) that the union of max{b,g} many families in the Baire space which are not finitely dominating is not dominating. In particular, it is consistent that for each nonprincipal ultrafilter U, the cofinality of the reduced ultrapower w^w/U is greater than max{b,g}. The model is constructed by oracle chain condition forcing, to which we give a self-contained introduction.

math.LO↗
arXiv2010

The combinatorics of tau-covers

We solve four out of the six open problems concerning critical cardinalities of topological diagonalization properties involving tau-covers, show that the remaining two cardinals are equal, and give a consistency result concerning this remaining cardinal. Consequently, 21 open problems concerning potential implications between these properties are settled. We also give structural results based on the combinatorial techniques.

math.GN↗
arXiv2004

Long low iterations

We try to control many cardinal characteristics by working with a notion of orthogonality between two families of forcings. We show that b^+<g is consistent

math.LO↗
arXiv2001

On needed reals

Following Blass, we call a real a ``needed'' for a binary relation R on the reals if in every R-adequate set we find an element from which a is Turing computable. We show that every real needed for Cof(N) is hyperarithmetic. Replacing ``R-adequate'' by ``R-adequate with minimal cardinality'' we get related notion of being ``weakly needed''. We show that is is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetical reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999.

math.LO↗