A solution to Frankl and Kupavskii's conjecture concerning Erdős-Kleitman matching problem

For integers $n\ge s\ge 2$, let $e(n,s)$ be the maximum size of a family $\mathcal{F}\subseteq 2^{[n]}$ with no $s$ pairwise disjoint members. The study of determining $e(n,s)$ is closely related to its uniform counterpart, the well-known Erdős matching conjecture. Frankl and Kupavskii conjectured an exact formula for $e((m+1)s-\mathrm{\ell },s)$ when $1\le \mathrm{\ell }\le \lceil s\mathrm{/}2\rceil$. We prove that for every fixed $m\ge 3$ and sufficiently large $s$, the extremal families for $e((m+1)s-\mathrm{\ell },s)$ are $P(m,s,\mathrm{\ell };L)\{A\subseteq [n]\text{\hspace{0.17em}⁣}:\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }A\cap L\mathrm{\mid }\ge m+1\}$ for some $L$ with $\mathrm{\mid }L\mathrm{\mid }=\mathrm{\ell }-1$ when $1\le \mathrm{\ell }\le (\frac{m+1}{2m+1}-o(1))s$. In particular, this confirms the Frankl–Kupavskii conjecture for every fixed $m\ge 3$ and all sufficiently large $s$. For $m=3$, we determine the whole range of $\mathrm{\ell }$ for which $P(3,s,\mathrm{\ell };L)$ is extremal, generalizing a theorem of Kupavskii and Sokolov.