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

## TL;DR

This paper advances extremal set theory by providing new proofs and resolving specific cases of the maximum size of families of subsets with no disjoint subfamilies of sizes 3 and 4, building on classical and recent results.

## Contribution

It offers a shorter proof for a known case and resolves a new case for the maximum size of set families avoiding disjoint subsets of sizes 3 and 4.

## Key findings

- Provided a shorter proof for Quinn's case s=3, n≡1 mod 3.
- Resolved the case s=4, n≡2 mod 4.
- Extended the understanding of extremal set families avoiding certain disjoint subfamilies.

## 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 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 $s\ge 5$.