Families of sets with no matchings of sizes 3 and 4

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 Erdős and resolved for n≡0,−1(mods) 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≡1(mod3) 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≡2(mod4). This complements the results in our recent paper, where, in particular, we answered the question in the case n≡−2(mods) for s≥5.