arXiv subjects

Osvaldo Guzman

Publications and source records attributed to Osvaldo Guzman.

8 recordsLinked to original sources

arXiv2026

Square-bracket operations clubs

This paper continues the investigation of the three square-bracket operations $[\cdot\cdot]$ from chapter 5 of \cite{Walks}. \ We say that a square-bracket operation $[\cdot\cdot]$ has the \emph{Ramsey club property} if for every club $C\subseteqω_{1}$, there is an uncountable subset $W$ $\subseteq ω_{1}$ such that $\left[ αβ\right] \in C$ for every $α,β\in W.$ \ The second author proved that the Proper Forcing Axiom\textsf{ }implies that all the square-bracket operations induced by Aronszajn trees have this property. We extend this result to the other two classes. We conclude that each of the statements \textquotedblleft all square-bracket operations have the Ramsey club property\textquotedblright\ and \textquotedblleft No square-bracket operation has the Ramsey club property\textquotedblright\ are consistent with \textsf{ZFC. }In other words, \textsf{ZFC }is unable to decide the status of the Ramsey club property for any square-bracket operation. Furthermore, we analyze the status of the Ramsey club property for square-bracket operations under Martin's Axiom and the Continuum Hypothesis.

math.LO↗

arXiv2025

Forcing Axioms and construction schemes

We continue the development of the theory of construction schemes over $ω_1$ as introduced by the third author by studying their relation with forcing axioms. Formally, we introduce the cardinals $\mathfrak{m}^n_{\mathcal{F}}$ and use the consistency of $\mathfrak{m}^2_\mathcal{F}>ω_1$ to prove a fundamental result relating gaps and almost disjoint families over $ω$. The cardinals $\mathfrak{m}_\mathcal{F}$ are also used to prove some limiting results for contstruction schemes, some of which answer questions from \cite{schemescruz}. Finally, we show that PID implies the non-existence of $2$-capturing schemes.

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arXiv2023

Infinite dimensional sequential compactness: Sequential compactness based on barriers

We introduce a generalization of sequential compactness using barriers on $ω$ extending naturally the notion introduced in [W. Kubiś and P. Szeptycki, On a topological Ramsey theorem, \emph{Canad. Math. Bull.}, 66 (2023), {156}--{165}]. We improve results from [C. Corral and O. Guzm{á}n and C. L{ó}pez-Callejas, High dimensional sequential compactness, \emph{Fund. Math.}] by building spaces that are $\mathcal{B}$-sequentially compact but no $\mathcal{C}$-sequentially compact when the barriers $\mathcal{B}$ and $\mathcal{C}$ satisfy certain rank assumption which turns out to be equivalent to a Katětov-order assumption. Such examples are constructed under the assumption $\mathfrak{b} =\mathfrak{c}$. We also exhibit some classes of spaces that are $\mathcal{B}$-sequentially compact for every barrier $\mathcal{B}$, including some classical classes of compact spaces from functional analysis, and as a byproduct we obtain some results on angelic spaces. Finally we introduce and compute some cardinal invariants naturally associated to barriers.

math.GN↗

arXiv2021

The ultrafilter and almost disjointness numbers

We prove that every MAD family can be destroyed by a proper forcing that preserves $P$-points. With this result, we prove that it is consistent that $ω_{1}=\mathfrak{u}<\mathfrak{a,}$ solving a nearly 20 year old problem of Shelah and a problem of Brendle. We will also present a simple proof of a result of Blass and Shelah that the inequality $\mathfrak{u<s}$ is consistent.

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arXiv2018

P-points, MAD families and Cardinal Invariants

This is the Ph.D. thesis of the author, which was written under the supervision of Michael Hrušák at UNAM. The main contributions of this thesis are the following: There is a $+$-Ramsey \textsf{MAD} family. This answers an old question of Michael Hrušák. There are no $P$-points in the Silver model, answering a question of Michael Hrušák (this is joint work with David Chodounský. The statement \textquotedblleft There are no $P$-points\textquotedblright\ is consistent with the continuum being arbitrarily large, this answers an open question regarding $P$-points. Every Miller indestructible \textsf{MAD} family is $+$-Ramsey. This improves a result of Hrušák and Garc\'ıa Ferreira. A Borel ideal is Shelah-Steprāns if and only if it is Katětov above \textsf{FIN}$\times$\textsf{FIN}$.$ This entails that Shelah-Steprāns \textsf{MAD} families have very strong indestructibility properties. Cohen indestructible \textsf{MAD} families exist generically if and only if $\mathfrak{b=c}$. The equality \textsf{non}$\left( \mathcal{M}\right) =ω_{1}$ implies the $\left( \ast\right) $ principle of Sierpiński. This answers a question of Arnie Miller.

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arXiv2018

On $\left( 1,ω_{1}\right) $\emph{-}weakly universal functions

A function $U:\left[ ω_{1}\right] ^{2}\longrightarrowω$ is called $\left( 1,ω_{1}\right) $\emph{-weakly universal }if for every function $F:\left[ ω_{1}\right] ^{2}\longrightarrowω$ there is an injective function $h:ω_{1}\longrightarrowω_{1}$ and a function $e:ω\longrightarrowω$ such that $F\left( α,β\right) =e\left( U\left( h\left( α\right) ,h\left( β\right) \right) \right) $ for every $α,β\inω_{1}$. We will prove that it is consistent that there are no $\left( 1,ω_{1}\right) $\emph{-}weakly universal functions, this answers a question of Shelah and Steprāns. In fact, we will prove that there are no $\left( 1,ω_{1}\right) $\emph{-}weakly universal functions in the Cohen model and after adding $ω_{2}$ Sacks reals side-by-side. However, we show that there are $\left( 1,ω_{1}\right) $\emph{-}weakly universal functions in the Sacks model. In particular, the existence of such graphs is consistent with $\clubsuit$ and the negation of the Continuum Hypothesis.

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arXiv2018

There is a $+$-Ramsey \textsf{MAD} family

We answer an old question of Michael Hrušák by constructing a $+$-Ramsey \textsf{MAD} family without the need of any additional axioms beyond $\mathsf{ZFC.}$ We also prove that every Miller-indestructible \textsf{MAD }family is $+$-Ramsey, this improves a result of Michael Hrušák.

math.LO↗

arXiv2017

How to drive our families mad

Given a family $F$ of pairwise almost disjoint sets on a countable set $S$, we study maximal almost disjoint (mad) families $F^+$ extending $F$. We define $a^+(F)$ to be the minimal possible cardinality of $F^+\setminus F$ for such $F^+$, and $a^+(κ)=\sup\{a^+(F): |F| \leq κ\}$. We show that all infinite cardinal less than or equal to the continuum continuum can be represented as $a^+(F)$ for some almost disjoint $F$ and that the inequalities $\aleph_1=a<a^+(\aleph_1)=c$ and $a=a^+(\aleph_1)<c$ are both consistent. We also give a several constructions of mad families with some additional properties.

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