Related papers: Fin-intersecting MAD families

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 show that all maximal almost disjoint families have pseudocompact Vietoris hyperspace if and only if $\mathsf{MA}_\mathfrak c (\mathcal P(\omega)/\mathrm{fin})$ holds. We further study the question whether there is a maximal almost…

We show that both $\mathfrak{ap=c}$ and $\diamondsuit(\mathfrak{b})$ imply the existence of MAD families with pseudocompact Vietoris hyperspace, substantially expanding the list of models where their existence is known. We also discuss some…

We explore almost-normality in Isbell-Mr\'owka spaces and some related concepts. We use forcing to provide an example of an almost-normal not normal almost disjoint family, explore the concept of semi-normality in Isbell-Mr\'owka spaces,…

Let $V$ be a finite dimensional vector space over a finite field, and $\mathcal{F}$ a family consisting of $k$-subspaces of $V$. The family $\mathcal{F}$ is called $t$-intersecting if $\dim(F_{1}\cap F_{2})\geq t$ for any $F_{1}, F_{2}\in…

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 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 $F^+$,…

A MAD (maximal almost disjoint) family is an infinite subset A of the infinite subsets of {0,1,2,..} such that any two elements of A intersect in a finite set and every infinite subset of {0.1.2...} meets some element of $\aa$ in an…

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 recall some classical results relating normality and some natural weakenings of normality in $\Psi$-spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$-sets, $\lambda$-sets and…

A family $\mathcal{F}$ of $k$-subsets of an $n$-set is called $s$-almost $t$-intersecting if each member is $t$-disjoint with at most $s$ members. In this paper, we prove that, if $\left|\mathcal{F}\right|$ is maximum, then $\mathcal{F}$…

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…

Let $\mathcal{F}$ be a family of $k$-dimensional subspaces of an $n$-dimensional vector space. Write $\mathcal{D}_{\mathcal{F}}(H;t)=\{F\in \mathcal{F}\colon \dim(F\cap H)\leq t \}$ for a subspace $H$. The family $\mathcal{F}$ is called…

An almost disjoint family $\mathcal A$ of subsets of $\mathbb N$ is said to be $\mathbb R$-embeddable if there is a function $f:\mathbb N\rightarrow \mathbb R$ such that the sets $f[A]$ are ranges of real sequences converging to distinct…

In this work, we introduce a natural notion concerning finite vector spaces. A family of $k$-dimensional subspaces of $\mathbb{F}_q^n$, which forms a partial spread, is called almost affinely disjoint if any $(k+1)$-dimensional subspace…

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of~$[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \emptyset$ for all $F, F' \in \mathcal F$. It is called almost…

Almost disjoint families of true cardinality $\mathfrak{c}$ are used to produce an example of a mildly-normal not partly-normal $\Psi$-space and a quasi-normal not almost-normal $\Psi$-space. This is related with a problem posed by Lufti…

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…

Let $V$ be an $(n+\ell)$-dimensional vector space over a finite field, and $W$ a fixed $\ell$-dimensional subspace of $V$. Write ${V\brack n,0}$ to be the set of all $n$-dimensional subspaces $U$ of $V$ satisfying $\dim(U\cap W)=0$. A…

We investigate families of subsets of $\omega$ with almost disjoint refinements in the classical case as well as with respect to given ideals on $\omega$. More precisely, we study the following topics and questions: 1) Examples of…