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arXiv · 2605.04379

More on the Erd\H os--Kleitman problem on matchings in set families

Andrey Kupavskii·Georgy Sokolov
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Abstract

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 Erd\H{o}s, 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 Erd\H{o}s 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 $s\ge s_0(m)$.

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Andrey Kupavskii, Georgy Sokolov. 2026-05-06. More on the Erd\H os--Kleitman problem on matchings in set families. https://arxiv.org/abs/2605.04379

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