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 $F^+$, and $a^+(κ)=\sup\{a^+(F): |F| \leq κ\}$. We show that all infinite cardinal less than or equal to the continuum continuum can be represented as $a^+(F)$ for some almost disjoint $F$ and that the inequalities $\aleph_1=a<a^+(\aleph_1)=c$ and $a=a^+(\aleph_1)<c$ are both consistent.
We also give a several constructions of mad families with some additional properties.
We prove the consistency of ZF+DC+"there are no mad families"+"there exists a non-meager filter on $ω$" relative to ZFC, answering a question of Neeman and Norwood. We also introduce a weaker version of madness, and we strengthen the result from [HwSh:1090] by showing that no such families exist in our model.
We consider the definability of mad families in vector spaces of the form $\underset{n<ω}{\bigoplus} F$ where $F$ is a field of cardinality $\leq \aleph_0$. We show that there is no analytic mad family of subspaces when $F=\mathbb{F}_2$, partially answering a question of Smythe. Our proof relies on a variant of Mathias forcing restricted to a certain idempotent ultrafilter whose existence follows from Glazer's proof of Hindman's theorem.
We show that if all collections of infinite subsets of $\N$ have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. This solves a long-standing problem going back to Mathias \cite{mathias}. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation $E_0$, and thus is constant on a "large" set. Furthermore we announce a number of additional results about mad families relativ…
We study the notion of $\mathcal J$-MAD families where $\mathcal J$ is a Borel ideal on $ω$. We show that if $\mathcal J$ is an arbitrary $F_σ$ ideal, or is any finite or countably iterated Fubini product of $F_σ$ ideals, then there are no analytic infinite $\mathcal J$-MAD families, and assuming Projective Determinacy there are no infinite projective $\mathcal J$-MAD families; and under the full Axiom of Determinacy + $V=\mathbf{L}(\mathbb{R})$ there are no infinite $\mathcal J$-mad families. These results apply in particular when $\mathcal J$ is the ideal of finite sets $\mathrm{Fin}$, which…
We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $κ$ is indestructible by the higher random forcing $\mathbb Q_κ$. We then use this characterisation to show that $\mathrm{add}(\mathbf{null}_κ) = \mathfrak b_κ= \mathfrak c_κ$ implies the existence $\mathbb Q_κ$-indestructible family. The results and proofs presented here are parallel to those for classical random forcing.
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 family or include a member of the family). We prove that it is hard to prove the consistency of the negation: (a) if 2^{aleph_0} < aleph_omega, then there is such a family (b) if there is no such families then some situation related to pcf holds whose consistency is large.
This is the Ph.D. thesis of the author, which was written under the supervision of Michael Hrušák at UNAM. The main contributions of this thesis are the following: There is a $+$-Ramsey \textsf{MAD} family. This answers an old question of Michael Hrušák. There are no $P$-points in the Silver model, answering a question of Michael Hrušák (this is joint work with David Chodounský. The statement \textquotedblleft There are no $P$-points\textquotedblright\ is consistent with the continuum being arbitrarily large, this answers an open question regarding $P$-points. Every Miller indestructible \text…
We consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and that the "spectrum" of cardinalities of mad families of subspaces can be made arbitrarily large, in analogy to results for mad families on $ω$ . We apply the author's local Ramsey theory for vector spaces to give partial results concerning their definability.
We prove in ZFC that there is a MAD family of functions in omega^omega which is also maximal with respect to infinite partial functions. This solves a 20 year old question of Van Douwen. We also strengthen a result of J. Steprans stating that strongly MAD families of functions cannot be analytic. We show that analytic MAD families of functions, if they exist, must satisfy some strong constraints.
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 witnessed by the Laver model for the consistency of Borel's conjecture.
We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $ω$ implies that the modal logic S4.1.2 is complete with respect to the Čech-Stone compactification of the natural numbers, the space $βω$. In the same fashion we prove that the modal logic S4 is complete with respect to the space $ω^*=βω\setminusω$. This improves the results of G. Bezhanishvili and J. Harding who prove these theorems under stronger assumptions ($\mathfrak{a}=\mathfrak{c}$). Our proof is also somewhat simpler.
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 orthogonal families, and that CH implies that for any very mad family there is one orthogonal to it. Finally we prove that the axiom of constructibility implies that there exists a coanalytic very mad family. Cofinitary groups have a natural action on the natural numbers. We prove that a maximal cofinitary group cannot have infinitely many orbits under this action…
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 almost disjoint family can be maximal, and show more generally that if Martin's Axiom holds at $κ<2^{\aleph_0}$, then no $κ$-Souslin infinite almost disjoint family can be maximal. Finally we show that if $\aleph_1^{L[a]}<\aleph_1$, then there are no $Σ^1_2[a]$ infinite mad families.
Two sets are said to be almost disjoint if their intersection is finite. Almost disjoint subsets of [omega]^omega and omega^omega have been studied for quite some time. In particular, the cardinal invariants a and a_e, defined to be the minimum cardinality of a maximal infinite almost disjoint family of [omega]^omega and omega^omega respectively, are known to be consistently less than continuum. Here we examine analogs for functions in R^omega and projections on l^2, showing that they too can be consistently less than continuum.
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 families on definable structures and of certain mad families (like those added by Hechler's poset for adding an a.d. family) regardless of the cofinality of their size. In particular, we define a class of posets called $σ$-Frechet-linked and show that they work well to preserve mad families, and unbounded families on $ω^ω$. As applications of this method, w…
The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on omega. We treat the problem to characterize those sets A which, in some forcing extension of the universe, can be the mad spectrum. We solve this problem to some extent. What remains open is the possible values of min(A) and max(A).
We present a variation of the proof in the first author's "Mob families and mad families" of Con(b<a), which in particular removes some of the obstacles to generalising the argument to uncountable cardinals.
We survey results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, $\mathcal I$-mad families, maximal eventually different families, and maximal cofinitary groups. We discuss the non-increasing sequence of cardinal characteristics $\mathfrak a_ξ$, for $ξ<ω_1$ as well as the notions of spectra of characteristics and optimal projective witnesses. We give an account of Zhang's forcing to add generic cofinitary permutations, and of a version of this forcing with built-in coding.
