Related papers: Maximal almost disjoint families, determinacy, and…

Starting from an inaccessible cardinal, we construct a model of $ZF+DC$ where there exists a mad family and all sets of reals are $\mathbb Q$-measurable for $\omega^{\omega}$-bounding sufficiently absolute forcing notions $\mathbb Q$.

We call a group FJ if it satisfies the $K$- and $L$-theoretic Farrell-Jones conjecture with coefficients in $\mathbb Z$. We show that if $G$ is FJ, then the simple Borel conjecture (in dimensions $\ge 5$) holds for every group of the form…

Let $E$ be a vector space over a countable field of dimension $\aleph_0$. Two infinite-dimensional subspaces $V,W \subseteq E$ are almost disjoint if $V \cap W$ is finite-dimensional. This paper provides some improvements on results about…

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 ${}^\omega \omega$-bounding, preserves…

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…

We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $\kappa$ is indestructible by the higher random forcing $\mathbb Q_\kappa$. We then use this characterisation to show that…

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…

Let J be a strongly stable monomial ideal in S=K[x_1,...,x_n] and let Mf(J) be the family of all homogeneous ideals I in S such that the set of all terms outside J is a K-vector basis of the quotient S/I. We show that an ideal I belongs to…

We prove that CH implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while $\mathfrak b=\mathfrak c$ is consistent with the negation of this statement as…

We continue the project of the study of reverse mathematics principles inspired by cardinal invariants. In this article in particular we focus on principles encapsulating the existence of large families of objects that are in some sense…

We propose a new, game-theoretic, approach to the idealized forcing, in terms of fusion games. This generalizes the classical approach to the Sacks and the Miller forcing. For definable ($\mathbf{\Pi}^1_1$ on $\mathbf{\Sigma}^1_1)…

Let $\mathcal R$ be a $\Sigma^1_1$ binary relation, and recall that a set $A$ is $\mathcal R$-discrete if no two elements of $A$ are related by $\mathcal R$. We show that in the Sacks and Miller forcing extensions of $L$ there is a…

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega$ implies that the modal logic S4.1.2 is complete with respect to the \v{C}ech-Stone compactification of the natural numbers,…

Let $\mathcal R$ be a $\Sigma^1_1$ binary relation and call a set $\mathcal R$-discrete iff no two distinct of its elements are $\mathcal R$-related. We show that in the extension of $\mathbf{L}$ by iterated Sacks forcing, there is a…

It is well-known that a finite axiomatization of Zermelo-Fraenkel set theory (ZF) is not possible in the same first-order language. In this note we show that a finite axiomatization is possible if we extent the language of ZF with the new…

In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if B is a G-delta-sigma set, then either B is countable or B contains a perfect subset. Second, we…

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…

We study families of subsets of $\omega$ which are independent with respect to the asymptotic density $\mathsf{d}$. We show, for instance, that there exists a maximal $\mathsf{d}$-independent family $\mathcal{A}$ such that…

The collection of branches (maximal linearly ordered sets of nodes) of the tree ${}^{<\omega}\omega$ (ordered by inclusion) forms an almost disjoint family (of sets of nodes). This family is not maximal -- for example, any level of the tree…

The problem we are considering came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants like the determinacy can be bounded simultaneously in families of formal…