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

Randić structure of a graph

AbstractLet G be a collection of graphs with n vertices. We present a simple description of [G]χ={H∈G:χ(H)=χ(G)} where χ denotes the Randić index. We associate to G a Q-linear map ρ:Qm→Qk (for some integers k,m depending on G) such that the kernel of ρ contains the necessary information to describe [G]χ in terms of linear equations. These results provide precise tools for analyzing the behavior of χ on a collection of graphs

On the Polishness of the inverse semigroup Γ(X)\Gamma(X) on a compact metric space XX

Let $\Gamma(X)$ be the inverse semigroup of partial homeomorphisms between open subsets of a compact metric space $X$. There is a topology, denoted $\tau_{hco}$, that makes $\Gamma(X)$ a topological inverse semigroup. We address the question of whether $\tau_{hco}$ is Polish. For a 0-dimensional compact metric space $X$, we prove that $(\Gamma(X), \tau_{hco})$ is Polish by showing that it is topologically isomorphic to a closed subsemigroup of the Polish symmetric inverse semigroup $I(\N)$. We present examples, similar to the classical Munn semigroups, of Polish inverse semigroups consisting of partial isomorphism on lattices of open 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