  In this paper, we study the following classical question of extremal set
theory: what is the maximum size of a family of subsets of $[n]$ such that no
$s$ sets from the family are pairwise disjoint? This problem was first posed by
Erd\H os and resolved for $n\equiv 0, -1\ (\mathrm{mod }\ s)$ by Kleitman in
the 60s. Very little progress was made on the problem until recently. The only
result was a very lengthy resolution of the case $s=3,\ n\equiv 1\ (\mathrm{mod
}\ 3)$ by Quinn, which was written in his PhD thesis and never published in a
refereed journal. In this paper, we give another, much shorter proof of Quinn's
result, as well as resolve the case $s=4,\ n\equiv 2\ (\mathrm{mod }\ 4)$. This
complements the results in our recent paper, where, in particular, we answered
the question in the case $n\equiv -2\ (\mathrm{mod }\ s)$ for $s\ge 5$.
