A solution to Frankl and Kupavskii's conjecture concerning Erd\H{o}s-Kleitman matching problem

For integers $n\ge s\ge2$, let $e(n,s)$ be the maximum size of a family $\mathcal F\subseteq2^{[n]}$ with no $s$ pairwise disjoint members. The study of determining $e(n,s)$ is closely related to its uniform counterpart, the well-known Erd\H{o}s matching conjecture. Frankl and Kupavskii conjectured an exact formula for $e((m+1)s-\ell,s)$ when $1\le \ell\le \lceil s/2\rceil$. We prove that for every fixed $m\ge3$ and sufficiently large $s$, the extremal families for $e((m+1)s-\ell,s)$ are $P(m,s,\ell;L)\coloneqq\{A\subseteq [n]\colon |A|+|A\cap L|\ge m+1\}$ for some $L$ with $|L|=\ell-1$ when $1\le \ell\le (\frac{m+1}{2m+1}-o(1))s$. In particular, this confirms the Frankl--Kupavskii conjecture for every fixed $m\ge3$ and all sufficiently large $s$. For $m=3$, we determine the whole range of $\ell$ for which $P(3,s,\ell;L)$ is extremal, generalizing a theorem of Kupavskii and Sokolov.