Extremal Families for the Erd\H{o}s--Kleitman Problem: The Missing Constructions

Abstract

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 problem of determining $e(n,s)$, now called the Erd\H{o}s--Kleitman problem, is closely related to the well-known Erd\H{o}s matching problem. Frankl and Kupavskii posed a meta-conjecture predicting that the maximum is always attained by a weighted family. Fix $m\ge3$, write $n=ms+c$ with $0\le c<s$, and set $\ell=s-c$. For $0\le k\le m$, let $a_k=ms-kc-1$. For $A\in\binom{[n]}{a_k}$, define \[ \mathcal H^k(m,s,\ell;A):= \{F\subseteq[n]: k|F|+|F\cap A|\ge m(k+1)\}. \] This defines a unified class of weighted families with matching number less than $s$. Among these families, $\mathcal H^0$, $\mathcal H^1$, and $\mathcal H^m$ were previously known to be extremal in different ranges of $c$. We show that for $1\le k\le m-1$, all families $\mathcal H^k$ are uniquely extremal in some ranges of $c$. More precisely, we prove that for every $m\ge3$ and every $1\le k\le m-1$, there exist constants $\alpha=\alpha(m,k)>0$, $\beta=\beta(m,k)>0$ and an integer $s_0=s_0(m,k)$ such that, for all integers $s\ge s_0$ and all integers $c$ with $0\le c<s$, the only extremal families for $e(n,s)$ are the families $\mathcal H^k(m,s,\ell;A)$ with $A\in\binom{[n]}{a_k}$ whenever $\beta s^{(k-1)/k}\le c\le \alpha s^{k/(k+1)}$. In particular, this result determines an infinite number of new extremal families for the Erd\H{o}s--Kleitman problem and verifies the Frankl--Kupavskii meta-conjecture in these ranges. This also provides a quantitative extension of the result of Kupavskii and Sokolov on the extremality of $\mathcal H^1$.