More on the Erd\H os--Kleitman problem on matchings in set families
Advances understanding of extremal set theory for the Erdős–Kleitman problem, but the result is approximate and conditional on large parameters.
The authors prove an approximate version of a conjecture by Frankl and Kupavskii on the maximum size of set families without s pairwise disjoint members, for large s relative to m.
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 $s\ge s_0(m)$.