P-points, MAD families and Cardinal Invariants

This is the Ph.D. thesis of the author, which was written under the supervision of Michael Hru\v{s}'{a}k at UNAM. The main contributions of this thesis are the following: There is a $+$-Ramsey $\mathsf{\text{MAD}}$ family. This answers an old question of Michael Hru\v{s}'{a}k. There are no $P$-points in the Silver model, answering a question of Michael Hru\v{s}'{a}k (this is joint work with David Chodounsk'{y}. 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 $\mathsf{\text{MAD}}$ family is $+$-Ramsey. This improves a result of Hru\v{s}'{a}k and Garc'{\i}a Ferreira. A Borel ideal is Shelah-Stepr={a}ns if and only if it is Kat\v{e}tov above $\mathsf{\text{FIN}}$$\times$$\mathsf{\text{FIN}}$$\mathrm{.}$ This entails that Shelah-Stepr={a}ns $\mathsf{\text{MAD}}$ families have very strong indestructibility properties. Cohen indestructible $\mathsf{\text{MAD}}$ families exist generically if and only if $\mathfrak{b}=\mathfrak{c}$. The equality $\mathsf{\text{non}}$$\left( $\mathcal{M}$\right) =\omega_{1}$impliesthe$\left( \ast\right) $ principle of Sierpi'{n}ski. This answers a question of Arnie Miller.