More on the Erd\H os--Kleitman problem on matchings in set families
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| Format: | Preprint |
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2026
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| _version_ | 1866917463275339776 |
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| author | Kupavskii, Andrey Sokolov, Georgy |
| author_facet | Kupavskii, Andrey Sokolov, Georgy |
| contents | 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 $s\ge s_0(m)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04379 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | More on the Erd\H os--Kleitman problem on matchings in set families Kupavskii, Andrey Sokolov, Georgy Combinatorics Discrete Mathematics 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 $s\ge s_0(m)$. |
| title | More on the Erd\H os--Kleitman problem on matchings in set families |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2605.04379 |