Related papers: On the bounding, splitting, and distributivity num…

Let $r_k(n)$ denote the maximum cardinality of a set $A \subset \{1,2, \dots, n \}$ such that $A$ does not contain a $k$-term arithmetic progression. In this paper, we give a method of constructing such a set and prove the lower bound…

We introduce a generalization of sequential compactness using barriers on $\omega$ extending naturally the notion introduced in [W. Kubi\'{s} and P. Szeptycki, On a topological Ramsey theorem, \emph{Canad. Math. Bull.}, 66 (2023),…

Combining stationary reflection (a compactness property) with the failure of SCH (an instance of non-compactness) has been a long-standing theme. We obtain this at $\aleph_{\omega_1}$, answering a question of Ben-Neria, Hayut, and Unger: We…

The object of the paper is to find complete systems of inequalities relating the perimeter $P$, the area $|\cdot|$ and the Cheeger constant $h$ of planar sets. To do so, we study the so called Blaschke--Santal\'o diagram of the triplet…

Let $\mathfrak{S}_n$ and $\mathfrak{B}_n$ denote the respective sets of ordinary and bigrassmannian (BG) permutations of order $n$, and let $(\mathfrak{S}_n,\leq)$ denote the Bruhat ordering permutation poset. We study the restricted poset…

Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles…

Let $\|\cdot\|$ denote the minimum distance to an integer. For $0<\gamma< 1$, $\theta>0$ and $(\alpha, \beta) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}$ we study when \begin{equation*} \|\alpha p^{\gamma}+\beta \|<p^{-\theta},…

We prove a lower bound of exp(-C (log(2/alpha))^7)N^{k-1} to the number of solutions of an invariant equation in k variables, contained in a set of density alpha. Moreover, we give a Behrend-type construction for the same problem with the…

In this note, we prove the existence of a set of $n$-fold Pfister forms of cardinality $2^n$ over $\mathbb{C}(x_1,\dots,x_n)$ which do not share a common $(n-1)$-fold factor. This gives a negative answer to a question raised by Becher. The…

We study several cardinal characteristics of closed graphs G on compact metrizable spaces. In particular, we address the question when it is consistent for the bounding number to be strictly smaller than the smallest size of a set not…

Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian…

We prove the consistency of $\binom{\mu^+}{\mu}\nrightarrow\binom{\mu^+ \omega_1}{\mu\ \mu}$ where $\mu$ is a strong limit singular cardinal of countable cofinality. This result can be forced at limit of measurable cardinals and at small…

We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr(B_0 times B_1)= max(irr(B_0),irr(B_1)). We prove…

We study the values of the higher dimensional cardinal characteristics for sets of functions $f:\omega^\omega \to \omega^\omega$ introduced by the second author. We prove that while the bounding numbers for these cardinals can be strictly…

Assume GCH and let $\lambda$ denote an uncountable cardinal. We prove that if $\square_\lambda$ holds, then this may be witnessed by a coherent sequence $< C_\alpha | \alpha < \lambda^+ >$ with the following remarkable guessing property:…

Let $M$ be a compact connected manifold of dimension $n$ endowed with a conformal class $C$ of Riemannian metrics of volume one. For any integer $k\geq0$, we consider the conformal invariant $\lambda_k ^c (C)$ defined as the supremum of the…

We present the classical theory of preservation of $\sqsubset$-unbounded families in generic extensions by ccc posets, where $\sqsubset$ is a definable relation of certain type on spaces of real numbers, typically associated with some…

If $\Gamma<\mathrm{PSL}(2,\mathbb{C})$ is a lattice, we define an invariant of a representation $\Gamma\rightarrow \mathrm{PSL}(n,\mathbb{C})$ using the Borel class $\beta(n)\in…

Under large cardinal hypotheses beyond the Kunen inconsistency -- hypotheses so strong as to contradict the Axiom of Choice -- we solve several variants of the generalized continuum problem and identify structural features of the levels…

We say that a family ${x_i|i\in[m]}$ of vectors in a Banach space $X$ satisfies the $k$-collapsing condition if $|\sum_{i\in I}x_i|\leq 1$ for all $k$-element subsets $I\subseteq{1,2,...,m}$. Let $C(k,d)$ denote the maximum cardinality of a…