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 Erdős 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.

翻译：给定整数$n\\ge s\\ge 2$，令$e(n,s)$表示一个$n$元集合的子集族的最大规模，该子集族不包含$s$个两两不交的成员。对这一量的研究可追溯至20世纪60年代，当时Kleitman对所有整数$m,s\\ge 1$确定了$e(sm-1,s)$和$e(sm,s)$的值。确定$e(n,s)$的问题与其均匀对应问题密切相关，后者是著名的Erdős匹配猜想的研究主题。确定$e(n,s)$的问题已被证明非常困难，尽管这些年间取得了一些进展，但甚至关于$e(n,s)$值的一般性猜想仍缺失。在本文中，我们完全解决了$n\\le 3s$情形下的问题。在此范围内，$s$-匹配中子集的平均大小至多为$3$，而'缺失的'$2$元和$3$元子集之间的微妙相互作用在此起着关键作用。在特征描述中出现了四类极值族。我们的结果揭示了极值函数$e(n,s)$在一般情形下可能的行为模式。