P-points, MAD families and Cardinal Invariants

Abstract

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 $\mathfrak{b}=\mathfrak{c}$. The equality \textsf{non}$\left(\mathcal{M}\right)={\omega }_1$ implies the $\left(\ast \right)$ principle of Sierpiński. This answers a question of Arnie Miller.

Ph.D. thesis, UNAM (2018), under supervision of Michael Hrušák