Countably compact groups without non-trivial convergent sequences

We construct, in $\mathsf{ZFC}$, a countably compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences, answering an old problem of van Douwen. As a consequence we also prove the existence of two countably compact groups $\mathbb{G}_{0}$ and $\mathbb{G}_{1}$ such that the product $\mathbb{G}_{0} \times \mathbb{G}_{1}$ is not countably compact, thus answering a classical problem of Comfort.Comment: 21 pages, to be published in Transactions of the American Mathematical Societ

On the structure of Borel ideals in-between the ideals \ED and \fin\otimes\fin in the Kat\v{e}tov order

For a family \cF\subseteq \omega^\omega we define the ideal \I(\cF) on $\omega\times\omega$ to be the ideal generated by the family \{A\subseteq \omega\times\omega:\exists f\in \cF\,\forall^\infty n\, (|\{k:(n,k)\in A\}|\leq f(n))\}. Using ideals of the form \I(\cF), we show that the structure of Borel ideals in-between two well known Borel ideals \ED = \{A\subseteq\omega\times\omega:\exists m \, \forall^\infty n\, (|\{k:(n,k)\in A\}| and \fin\otimes\fin = \{A\subseteq\omega\times\omega:\forall^\infty n \, (|\{k:(n,k)\in A\}|<\aleph_0))\} in the Kat\v{e}tov order is fairly complicated. Namely, there is a copy of \cP(\omega)/\fin in-between \ED and \fin\otimes\fin, and consequently there are increasing and decreasing chains of length \bnumber and antichains of size \continuum

Reconstruction of a coloring from its homogeneous sets

We study a reconstruction problem for colorings. Given a finite or countable set $X$, a coloring on $X$ is a function $\varphi: [X]^{2}\to \{0,1\}$, where $[X]^{2}$ is the collection of all 2-elements subsets of $X$. A set $H\subseteq X$ is homogeneous for $\varphi$ when $\varphi$ is constant on $[H]^2$. Let $hom(\varphi)$ be the collection of all homogeneous sets for $\varphi$. The coloring $1-\varphi$ is called the complement of $\varphi$. We say that $\varphi$ is {\em reconstructible} up to complementation from its homogeneous sets, if for any coloring $\psi$ on $X$ such that $hom(\varphi)=hom(\psi)$ we have that either $\psi=\varphi$ or $\psi=1-\varphi$. We present several conditions for reconstructibility and non reconstructibility. We show that there is a Borel way to reconstruct a coloring from its homogeneous sets

Pathology of submeasures and FσF_\sigma ideals

We address some phenomena about the interaction between lower semicontinuous submeasures on $\mathbb{N}$ and $F_{\sigma}$ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological $F_\sigma$ ideals. We give a partial answers to the question of whether every nonpathological tall $F_\sigma$ ideal is Kat\v{e}tov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological $F_\sigma$ ideals using sequences in Banach spaces.Comment: arXiv admin note: substantial text overlap with arXiv:2111.1059

Ideals on countable sets: a survey with questions

An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We present a survey of results about ideals on countable sets and include many open questions

Ideals on countable sets: a survey with questions

An ideal on a set X is a collection of subsets of X closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We present a survey of results about ideals on countable sets and include many open questions