A complete solution of the Erd\H{o}s-Kleitman matching problem for $n\le 3s$

TL;DR

This paper completely determines the maximum size of set families with no s pairwise disjoint members for all n up to 3s, revealing the structure of extremal families and advancing understanding of the Erdős Matching Conjecture.

Contribution

It provides a complete solution for e(n,s) when n ≤ 3s, identifying four types of extremal families and offering insights into the behavior of the extremal function.

Findings

Determined e(n,s) for n ≤ 3s.

Identified four types of extremal families.

Enhanced understanding of the Erdős Matching Conjecture.

Abstract

Given integers $n\ge s\ge 2$, let $e(n,s)$ stand for the maximum size of a family of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. The study of this quantity goes back to the 1960s, when Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all integer $m,s\ge 1$. The question of determining $e(n,s)$ is closely connected to its uniform counterpart, the subject of the famous Erd\H{o}s Matching Conjecture. The problem of determining $e(n,s)$ has proven to be very hard and, in spite of some progress during these years, even a general conjecture concerning the value of $e(n,s)$ is missing. In this paper, we completely solve the problem for $n\le 3s$. In this regime, the average size of a set in an $s$-matching is at most $3$, and it is a delicate interplay between the `missing' $2$- and $3$-element sets that plays a key role here. Four types of extremal families appear in the characterization. Our result sheds light on how the extremal function $e(n,s)$ may behave in general.