Related papers: Constructing subsets of a given packing index in A…

For a finite group $G$, we associate the quantity $\beta(G)=\frac{|L(G)|}{|G|}$, where $L(G)$ is the subgroup lattice of $G$. Different properties and problems related to this ratio are studied throughout the paper. We determine the second…

If $A$ is an infinite Boolean algebra the cardinal invariant $\mathfrak{a}(A)$ is defined as the smallest size of an infinite partition of $A$. The cardinal $\mathfrak{a}(A\oplus B)$, where $A\oplus B$ is the free product of the Boolean…

We present a wide class of reflexive, precompact, non-compact, Abelian topological groups $G$ determined by three requirements. They must have the Baire property, satisfy the \textit{open refinement condition}, and contain no infinite…

We use assembly maps to study $\mathbf{TC}(\mathbb{A}[G];p)$, the topological cyclic homology at a prime $p$ of the group algebra of a discrete group $G$ with coefficients in a connective ring spectrum $\mathbb{A}$. For any finite group, we…

We establish some general principles and find some counter-examples concerning the Pontryagin reflexivity of precompact groups and P-groups. We prove in particular that: (1) A precompact Abelian group G of bounded order is reflexive iff the…

We introduce the following weak version of Hadwiger's conjecture: If $G$ is a graph and $\kappa$ is a cardinal such that there is no coloring map $c:G \to \kappa$, then $K_\kappa$ is a minor of $G$. We prove that this statement is true for…

Generalizing results from \cite{DTk,DU} we study the fine structure of locally minimal (locally) precompact Abelian groups (these are the locally essential subgroups $G$ of LCA groups $L$, i.e., such that $G$ non-trivially meets all…

A set $P$ of vertices in a graph $G$ is an open packing if no two distinct vertices in $P$ have a common neighbor. Among all maximal open packings in $G$, the smallest cardinality is denoted $\rho^{\rm o}_L(G)$ and the largest cardinality…

Let $G$ be a $\sigma$-finite abelian group, i.e. $G=\bigcup_{n\geq 1} G_n$ where $(G_n)_{n\geq 1}$ is a non decreasing sequence of finite subgroups. For any $A\subset G$, let $\underline{\mathrm{d}}(A):=\liminf_{n\to\infty}\frac{|A\cap…

An inaccessible cardinal $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa.$ We prove that if there is a model of $\ZFC$ with two supercompact cardinals, then there is a model of \ZFC where…

A group $G$ is said to be factorized into subsets $A_1, A_2, \ldots, A_s\subseteq G$ if every element $g$ in $G$ can be uniquely represented as $g=g_1g_2\ldots g_s$, where $g_i\in A_i$, $i=1,2,\ldots,s$. We consider the following…

The main result of this paper is to show that, if $\kappa$ is the smallest real-valued measurable cardinal not greater than $ 2^{\aleph_0}$, then there exists a complete metric space of cardinality not greater than $ 2^{\kappa}$ admitting a…

Let T be a complete, first-order theory in a finite or countable language having infinite models. Let I(T,kappa) be the number of isomorphism types of models of T of cardinality \kappa. We denote by \mu (respectively \hat\mu) the number of…

Many open conjectures in the representation theory of finite groups can be studied by reducing them to related questions about quasi-simple groups. In such studies, $p$-radical subgroups typically play a critical role. To classify the…

A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and…

(withdrawn.) For every lambda we give an explicit construction of an Abelian group with no non-trivial automorphisms. In particular the group absolutely has no non-trivial automorphisms, hence is absolutely indecomposable. Earlier we knew a…

Let $\{G_i :i\in\N\}$ be a family of finite Abelian groups. We say that a subgroup $G\leq \prod\limits_{i\in \N}G_i$ is \emph{order controllable} if for every $i\in \mathbb{N}$ there is $n_i\in \mathbb{N}$ such that for each $c\in G$, there…

In this article, we characterise geometrically when a right-angled Artin group splits over an abelian subgroup. More precisely, given a finite graph $\Gamma$, we show that $A(\Gamma)$ splits over an abelian subgroup if and only if it is…

For $n<\omega$, we say that the $\Pi^1_n$-reflection principle holds at $\kappa$ and write $\text{Refl}_n(\kappa)$ if and only if $\kappa$ is a $\Pi^1_n$-indescribable cardinal and every $\Pi^1_n$-indescribable subset of $\kappa$ has a…

On every set A there is a rigid binary relation i.e. such a relation R \subseteq A \times A that there is no homomorphism (A,R) \rightarrow (A,R) except the identity (Vop{\v{e}}nka et al. [1965]). We prove that for each infinite cardinal…