Families of sets with no matchings of sizes 3 and 4.

Abstract 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 (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 ≡ 1 (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 ≡ 2 (mod 4). This complements the results in our recent paper, where, in particular, we answered the question in the case n ≡ − 2 (mod s) for s ≥ 5. [ABSTRACT FROM AUTHOR]