Related papers: A solution to Roitman's problem

We prove that every MAD family can be destroyed by a proper forcing that preserves $P$-points. With this result, we prove that it is consistent that $\omega_{1}=\mathfrak{u}<\mathfrak{a,}$ solving a nearly 20 year old problem of Shelah and…

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 $2^{<\omega}$. Using a countable support…

Given a family $F$ of pairwise almost disjoint sets on a countable set $S$, we study maximal almost disjoint (mad) families $F^+$ extending $F$. We define $a^+(F)$ to be the minimal possible cardinality of $F^+\setminus F$ for such $F^+$,…

We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality $\mathfrak a_{\text{g}}$ of a maximal cofinitary group (MCG) is strictly between $\aleph_1$ and…

We show that, consistently, every MAD family has cardinality strictly bigger than the dominating number, that is a > d, thus solving one of the oldest problems on cardinal invariants of the continuum. The method is a contribution to the…

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} <…

We show that there are no infinite maximal almost disjoint ("mad") families in Solovay's model, thus solving a long-standing problem posed by A.D.R. Mathias in 1967. We also give a new proof of Mathias' theorem that no analytic infinite…

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 $\mathbb Q$.

A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family $\mathcal F$ of subsets of $[n]$ with all pairwise intersections of size $\lambda$ can have at most $n$ non-empty sets. One may weaken the…

We study two different types of (maximal) almost disjoint families: very mad families and (maximal) cofinitary groups. For the very mad families we prove the basic existence results. We prove that MA implies there exist many pairwise…

Given a countable transitive model $M$ for ZFC+CH, we prove that one can produce a maximal almost disjoint family in $M$ whose Vietoris Hyperspace of its Isbell-Mr\'owka space is pseudocompact on every Cohen extension of $M$. We also show…

We prove that if there is a dominating family of size ${\aleph}_{1}$, then there is are ${\aleph}_{1}$ many compact subsets of ${\omega}^{\omega}$ whose union is a maximal almost disjoint family of functions that is also maximal with…

We give the first (ZFC) dividing line in Keisler's order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal $\lambda$ for which there is $\mu < \lambda \leq 2^\mu$, we construct…

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…

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense,…

We throw some light on the question: is there a MAD family (= a family of infinite subsets of N, the intersection of any two is finite) which is completely separable (i.e. any X subseteq N is included in a finite union of members of the…

Let $\mathcal{F}$ be a family of subsets of $[n]$. The diameter of $\mathcal{F}$ is the maximum size of symmetric differences among pairs of its members. Resolving a conjecture of Erd\H{o}s, Kleitman determined the maximum size of a family…

We use coherent systems of FS iterations on a power set, which can be seen as matrix iteration that allows restriction on arbitrary subsets of the vertical component, to prove general theorems about preservation of certain type of unbounded…

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…

Assuming that every set is constructible, we find a $\Pi^1_1$ 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…