arXiv subjects

Barnabás Farkas

Publications and source records attributed to Barnabás Farkas.

7 recordsLinked to original sources

arXiv2026

Geometric duality, perfect graphs, and the Sierpiński space

In their classical paper \emph{On the stopping time Banach space}, Bang and Odell, among a plethora of results concerning the dyadic stopping time space and its dual, presented the first non-trivial example of the \emph{duality phenomenon} between combinatorial Banach spaces. We give a full characterization of such pairs $(\mc{F}_0, \mc{F}_1)$ of families of finite sets: This duality holds iff there is a perfect graph $G$ on $\NN$ such that $\mc{F}_0$ consists of all finite cliques of $G$ and $\mc{F}_1$ consists of all finite anti-cliques of $G$. As it turns out, Lovász' famous perfect graph theorem is an immediate corollary of this result. Among the many examples of such pairs of families, we investigate a particularly interesting one, when $G$ is the Sierpiński graph, and study general methods of embedding combinatorial and classical sequence spaces in the generated space, including the Schreier and $\ell_p$ spaces.

math.FA↗

arXiv2025

More on Halfway New Cardinal Characteristics

We continue investigating variants of the splitting and reaping numbers introduced in arXiv:1808.02442. In particular, answering a question raised there, we prove the consistency of $\mathrm{cof}(\mathcal{M})<\mathfrak{s}_{\frac{1}{2}}$ and of $\mathfrak{r}_{\frac{1}{2}}<\mathrm{add}(\mathcal{M})$. Moreover, we discuss their natural generalisations $\mathfrak{s}_ρ$ and $\mathfrak{r}_ρ$ for $ρ\in (0,1)$, and show that $\mathfrak{r}_ρ$ does not depend on $ρ$.

math.LO↗

arXiv2024

The ZOO of combinatorial Banach spaces

We study Banach spaces induced by families of finite sets in the most natural (Schreier-like) way, that is, we consider the completion $X_\mc{F}$ of $c_{00}$ with respect to the norm $\sup\{\sum_{k\in F}|x(k)|:F\in\mc{F}\}$ where $\mc{F}$ is an arbitrary (not necessarily compact) family of finite sets covering $\mbb{N}$. Among other results, we discuss the following: (1) Structure theorems bonding the combinatorics of $\mc{F}$ and the geometry of $X_\mc{F}$ including possible characterizations and variants of the Schur property, $\ell_1$-saturation, and the lack of copies of $c_0$ in $X_\mc{F}$. (2) A plethora of examples including a relatively simple $\ell_1$-saturated combinatorial space which does not satisfy the Schur property, as well as a new presentation of Pełczyński's universal space. (3) The complexity of the family $\{H\subseteq\NN:X_{\mc{F}\upharpoonright H}$ does not contain $c_0\}$.

math.FA↗

arXiv2019

Analytic P-ideals and Banach spaces

We study the interplay between Banach space theory and theory of analytic P-ideals. Applying the observation that, up to isomorphism, all Banach spaces with unconditional bases can be constructed in a way very similar to the construction of analytic P-ideals from submeasures, we point out numerous symmetries between the two theories. Also, we investigate a special case, the interactions between combinatorics of families of finite sets, topological properties of the "Schreier type" Banach spaces associated to these families, and the complexity of ideals generated by the canonical bases in these spaces. Among other results, we present some new examples of Banach spaces and analytic P-ideals, a new characterization of precompact families and its applications to enhance Pták's Lemma and Mazur's Lemma.

math.LO↗

arXiv2016

Towers in filters, cardinal invariants, and Luzin type families

We investigate which filters on $ω$ can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in $[ω]^ω$). We prove the following results: - Many classical examples of nice tall filters contain no towers (in ZFC). - It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well). - It is consistent that all towers generate non-meager filters, in particular (consistently) Borel filters do not contain towers. - The statement "Every ultrafilter contains towers." is independent of ZFC. Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters ($\mbox{add}^*(\mathcal F)$, $\mbox{cof}^*(\mathcal F)$, $\mbox{non}^*(\mathcal F)$, and $\mbox{cov}^*(\mathcal F)$), and the existence of Luzin type families (of size $\geq ω_2$), that is, if $\mathcal F$ is a filter then $X\subseteq [ω]^ω$ is an $\mathcal F$-Luzin family if $\{A\in X:|A\setminus F|=ω\}$ is countable for every $F\in \mathcal F$.

math.LO↗

arXiv2015

Almost disjoint refinements and mixing reals

We investigate families of subsets of $ω$ with almost disjoint refinements in the classical case as well as with respect to given ideals on $ω$. More precisely, we study the following topics and questions: 1) Examples of projective ideals. 2) We prove the following generalization of a result due to J. Brendle: If $V\subseteq W$ are transitive models, $ω_1^W\subseteq V$, $\mathcal{P}(ω)\cap V\not = \mathcal{P}(ω)\cap W$, and $\mathcal{I}$ is an analytic or coanalytic ideal coded in $V$, then there is an $\mathcal{I}$-almost disjoint refinement ($\mathcal{I}$-ADR) of $\mathcal{I}^+\cap V$ in $W$, that is, a family $\{A_X:X\in\mathcal{I}^+\cap V\}\in W$ such that (i) $A_X\subseteq X$, $A_X\in \mathcal{I}^+$ for every $X$ and (ii) $A_X\cap A_Y\in\mathcal{I}$ for every distinct $X$ and $Y$. 3) The existence of perfect $\mathcal{I}$-almost disjoint ($\mathcal{I}$-AD) families, and the existence of a "nice" ideal $\mathcal{I}$ on $ω$ with the property: Every $\mathcal{I}$-AD family is countable but $\mathcal{I}$ is nowhere maximal. 4) The existence of $(\mathcal{I},\text{Fin})$-almost disjoint refinements of families of $\mathcal{I}$-positive sets in the case of everywhere meager (e.g. analytic or coanalytic) ideals. We prove a positive result under Martin's Axiom. 5) Connections between classical properties of forcing notions and adding mixing reals (and mixing injections), that is, a (one-to-one) function $f:ω\toω$ such that $|f[X]\cap Y|=ω$ for every $X,Y\in [ω]^ω\cap V$.

math.LO↗

arXiv2010

More on cardinal invariants of analytic P-ideals

Given an ideal $I$ on $ω$ let $a(I) $ ($\bar{a}(I)$) be minimum of the cardinalities of infinite (uncountable) maximal $I$-almost disjoint subsets of $[ω]^ω$, and denote $b_I$ and$d_I$ the unbounding and dominating numbers of $(ω^ω,\le_I)$. We show that (1) $a(I)>omega$ if $I$ is a summable ideal; (2) $a(Z)=ω$ and $\bar{a}(Z)\le a$ if $Z$ is a tall density ideal, (3) $b\le \bar{a}(I)$, and $b_I=b$ and $d_I=d$, for any analytic P-ideal $I$ on $ω$. Given an analytic $P$-ideal $I$ we investigate the relationship between the Sack, the $I$-bounding, $I$-dominating and $ω^ω$-bounding properties of a given poset $P$. For example, for the density zero ideal $Z$ we can prove: (i) a poset $P$ is $Z$-bounding iff it has the Sacks property, (ii) if $P$ adds a slalom capturing all ground model reals then $P$ is $Z$-dominating.

math.LO↗