Set Theory Talks

Place: Fields Institute (Library)
Date: September 7, 2018 (13:30-15:00)
Speaker: Osvaldo Guzman
Title: On Restricted MADness
Abstract: Let $\mathcal{I}$ be an ideal on $\omega.$ We define \textsf{cov}$^{\ast}\left( \mathcal{I}\right) $ as the least size of a family
$\mathcal{B\subseteq I}$ such that for every infinite $X\in\mathcal{I}$ there is $B\in\mathcal{B}$ for which $B\cap X$ is infinite. We say an \textsf{AD} family $\mathcal{A\subseteq I}$ is a \emph{\textsf{MAD} family restricted to }$\mathcal{I}$ if for every infinite $X\in\mathcal{I}$ there is $A\in \mathcal{A}$ such that $\left\vert X\cap A\right\vert =\omega.$ The cardinal invariant $\mathfrak{a}\left( \mathcal{I}\right) $ is defined as the least size of an infinite \textsf{MAD} family restricted to $\mathcal{I}.$ The cardinal invariants $\mathfrak{o}$ and $\mathfrak{a}_{s}$ may be seen as
particular cases of this class of invariants. In this talk, we will prove that
if the maximum of $\mathfrak{a}$ and  textsf{cov}$^{\ast}\left(\mathcal{I}\right) $ is $\omega_{1}$ then $\mathfrak{a}\left( \mathcal{I}%
\right) =\omega_{1}.$ We will obtain some corollaries of this result. This is part of a joint work with Michael Hru\v{s}\'{a}k and Osvaldo Tellez.