arXiv subjects

Márcio Telles

Publications and source records attributed to Márcio Telles.

3 recordsLinked to original sources

arXiv2024

Density cardinals

How many permutations are needed so that every infinite-coinfinite set of natural numbers with asymptotic density can be rearranged to no longer have the same density? We prove that the density number $\mathfrak{dd}$, which answers this question, is equal to the least size of a non-meager set of reals, $\mathsf{non} (\mathcal{M})$. The same argument shows that a slight modification of the rearrangement number $\mathfrak{rr}$ of~\cite{BBBHHL20} is equal to $\mathsf{non} (\mathcal{M})$, and similarly for a cardinal invariant related to large-scale topology introduced by Banakh~\cite{Ba23}, thus answering a question of the latter. We then consider variants of $\mathfrak{dd}$ given by restricting the possible densities of the original set and / or of the permuted set, providing lower and upper bounds for these cardinals and proving consistency of strict inequalities. We finally look at cardinals defined in terms of relative density and of asymptotic mean, and relate them to the rearrangement numbers of~\cite{BBBHHL20}.

math.LO↗

arXiv2016

Spaces of completions of elementary theories and convergence laws for random hypergraphs

Consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p:\mathbb{N}\to[0,1]$. We prove that, for all logarithmo-exponential $p\ll n^{-d+ε}$, the probabilities of all elementary properties of hypergraphs converge, with particular emphasis in the ranges $p(n)\sim C/n^d$ and $p(n) \sim C\log(n)/n^d$. The exposition is unified by constructing, for each such function $p$, the topological space of all completions of its almost sure theory. This space turns out to be compact, metrizable and totally disconnected, but further properties depend on the range of $p$. The convergence of the probabilities of elementary properties is associated with a borelian probability measure on the space.

math.CO↗

arXiv2015

Some examples of asymptotic combinatorial behavior, zero-one and convergence results on random hypergraphs

This is an extended version of the thesis presented to the Programa de Pós-Graduação em Matemática of the Departamento de Matemática, PUC-Rio, in September 2013, incorporating some suggestions from the examining commission. Random graphs (and more generally hypergraphs) have been extensively studied, including their first order logic. In this work we focus on certain specific aspects of this vast theory. We consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p=p(n)$. We are particularly interested in the range $p(n) \sim C\log(n)/n^d$, after the double jump and near connectivity. We prove several zero-one, and, more generally, convergence results and obtain combinatorial applications of some

math.CO↗