Let $e(n,s)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. In 1968, answering a question of Erdős, Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all integers $m,s\ge 1$. Half a century later, Frankl and Kupavskii determined $e(s(m+1)-\ell, s)$ for $\ell \leq \frac{s-3}{m+3}$. They showed that the corresponding extremal example is closely connected with the extremal example for the Erdős Matching Conjecture, and conjectured that the same remains true for all $\ell \leq s/2$. In this paper, we prove an approximate version of their conjecture for $s\ge s_0(m)$.

翻译：设$e(n,s)$表示一个$n$元集合的子集族$\mathcal{F}$的最大可能大小，其中该族不包含$s$个两两不交的成员。1968年，Kleitman回答了Erdős提出的一个问题，确定了所有整数$m,s\ge 1$时的$e(sm-1,s)$和$e(sm,s)$。半个世纪后，Frankl与Kupavskii确定了当$\ell \leq \frac{s-3}{m+3}$时的$e(s(m+1)-\ell, s)$。他们表明相应的极值例子与Erdős匹配猜想的极值例子密切相关，并猜想对于所有$\ell \leq s/2$该结论仍然成立。本文中，我们证明了当$s\ge s_0(m)$时该猜想的一个近似版本。