Related papers: Bounded sequences having an even number of accumul…

We give a broad survey of inequalities for the number of linear extensions of finite posets. We review many examples, discuss open problems, and present recent results on the subject. We emphasize the bounds, the equality conditions of the…

We study M-separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b=d implies that the class of selectively separable spaces is not closed under finite products,…

A review of the state of the art of the comparison between any two different modes of convergence of sequences of measurable functions is carried out with focus on the algebraic structure of the families under analysis. As a complement of…

A set of sequences is said to converge simultaneously if there exists an infinite subset $H$ of the index set $\omega$ such that all sequences converge when restricted to $H$. We discuss simultaneous convergence of sequences in the same or…

The study of low regularity (in-)extendibility of Lorentzian manifolds is motivated by the question whether a given solution to the Einstein equations can be extended (or is maximal) as a weak solution. In this paper we show that a timelike…

We study supersolvable line arrangements in ${\mathbb P}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil)…

We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show, in particular, the relative consistency with GCH of the statement that every Banach space of density $\aleph_\omega$ contains…

In 1931, Banach proved that, far from being exceptional objects, the Weierstrass functions form a residual set in the space $\mathcal{C}[0,1]$ of continuous functions. Later on, in 1966, V. I. Gurariy showed that, except for zero, there is…

Assuming the generalized continuum hypothesis we construct arbitrarily big indecomposable Banach spaces. i.e., such that whenever they are decomposed as $X\oplus Y$, then one of the closed subspaces $X$ or $Y$ must be finite dimensional. It…

We study the link between stably finiteness and stably projectionless-ness for $C^*$-algebras of solvable Lie groups. We show that these two properties are equivalent if the dimension of the group is not divisible by $4$; otherwise, they…

In this note we extend a recent result of S. Brendle [3] to Riemannian manifolds with densities and nonnegative Bakry-\'Emery Ricci curvature.

The aim of this paper is to show a peculiar behavior of a (hypothetical) Collatz sequence going to infinity. We study the associated Syracusa sequence (the odd elements of the former) and show that the limit set of a conveniently normalized…

A recent result of Freeman, Odell, Sari, and Zheng states that whenever a separable Banach space not containing $\ell_1$ has the property that all asymptotic models generated by weakly null sequences are equivalent to the unit vector basis…

In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found,…

According to a classical result of Szemer\'{e}di, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson says that…

We isolate conditions on the relative size of sets of natural numbers $A,B$ that guarantee a nonempty intersection $\Delta(A)\cap\Delta(B)\ne\emptyset$ of the corresponding sets of distances. Such conditions apply to a large class of zero…

We study Hilbert generated versions of nonseparable Banach spaces $\mathcal X$ considered by Shelah, Stepr\=ans and Wark where the behavior of the norm on nonseparable subsets is so irregular that it does not allow any linear bounded…

In this note we generalize a criterion within the concept of infinite dense lineability due to Calder\'on-Moreno, Gerlach-Mena and Prado-Bassas. We also introduce and explore some local notions of lineability.

Using the classical technique of condensation of singularities, we prove that, for every zero-dimensional, complete separable metric space $G$, there exists a Suslinian, chainable metric continuum whose set of end points is homeomorphic to…

We present some new examples of separable (\mathcal_\infty) spaces which are (\ell_r) saturated for some (1 < r < \infty).