In this paper we study several kinds of maximal almost disjoint families. In the main result of this paper we show that for successor cardinals $\kappa$, there is an unexpected connection between invariants $\frac{a}_{e}(\kappa), \frac{b} (\kappa)$ and a certain cardinal invariant $\frac{m}_{d}(\kappa^{+})$ on $\kappa^{+}$. As a corollary we get for example the following result. For a successor cardinal $\kappa$, even assuming that $\kappa^{<\kappa}=\kappa$ and $2^{\kappa}=\kappa^{+}$, the following is not provable in Zermelo–Fraenkel set theory. There is a $\kappa^{+}$-cc poset which does not collapse $\kappa$ and which forces $\frac{a} (\kappa)=\kappa^{+}<\frac{a}_{e}(\kappa)=\kappa^{++}=2^{\kappa}$. We also apply the ideas from the proofs of these results to study $\frac{a} =\frac{a} (\omega)$ and $\hbox{\rm non}(\cal{M})$.
