  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 \textsf{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 \textsf{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
\textsf{FIN}$\times$\textsf{FIN}$.$ This entails that Shelah-Stepr\={a}ns
\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}$\left( \mathcal{M}\right)
=\omega_{1}$ implies the $\left( \ast\right) $ principle of Sierpi\'{n}ski.
This answers a question of Arnie Miller.
