arXiv subjects
Publications and source records attributed to Lukas Schembecker.
A maximal eventually different family is called $n$-productive if the product family $\mathcal{F}^n$ is still maximal. We construct closed $n$-productive families separating these strengthenings of maximality at every $n \geq 1$. Furthermore, we show how to force and construct an even stronger type of $\mathcal{I}_0$-productive family and discuss the relation of productivity to Van Douwen families.
For Van Douwen families, maximal families of eventually different permutations and maximal ideal independent families we show that the existence of a $Σ^1_2$ family implies the existence of a $Π^1_1$ family of the same size. We also prove a similar, but slightly weaker result for generating sets of cofinitary groups.
Kastermans proved that consistently $\bigoplus_{\aleph_1} \mathbb{Z}_2$ has a cofinitary representation. We present a short proof that $\bigoplus_{\mathfrak{c}} \mathbb{Z}_2$ always has an arithmetic cofinitary representation. Further, for every finite group $F$ we construct an arithmetic maximal cofinitary group of isomorphism type $(\ast_{\mathfrak{c}} \mathbb{Z}) \times F$. This answers an implicit question by Schrittesser and Mejak whether one may construct definable maximal cofinitary groups not decomposing into free products.
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.
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 $Δ^1_3$-wellorder of the reals and a co-analytic witness for $\mathfrak{a}_g$.
We prove that the spectrum of Van Douwen families is closed under singular limits. For any maximal eventually different family Raghavan defined in an associated ideal which measures how far the family is from being Van Douwen. Under CH we prove that every ideal containing Fin is realized as the associated ideal of some maximal eventually different family. Finally, we construct maximal eventually different families with Sacks-indestructible associated ideals to prove that in the iterated Sacks-model every $\aleph_1$-generated ideal containing Fin is realized.
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 $\text{CH}$.
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.