A complete solution of the Erdős-Kleitman matching problem for $n\le 3s$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914171970387968 |
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| author | Kupavskii, Andrey Sokolov, Georgy |
| author_facet | Kupavskii, Andrey Sokolov, Georgy |
| contents | Given integers $n\ge s\ge 2$, let $e(n,s)$ stand for the maximum size of a family of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. The study of this quantity goes back to the 1960s, when Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all integer $m,s\ge 1$. The question of determining $e(n,s)$ is closely connected to its uniform counterpart, the subject of the famous Erdős Matching Conjecture.
The problem of determining $e(n,s)$ has proven to be very hard and, in spite of some progress during these years, even a general conjecture concerning the value of $e(n,s)$ is missing. In this paper, we completely solve the problem for $n\le 3s$. In this regime, the average size of a set in an $s$-matching is at most $3$, and it is a delicate interplay between the `missing' $2$- and $3$-element sets that plays a key role here. Four types of extremal families appear in the characterization. Our result sheds light on how the extremal function $e(n,s)$ may behave in general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21628 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A complete solution of the Erdős-Kleitman matching problem for $n\le 3s$ Kupavskii, Andrey Sokolov, Georgy Combinatorics Discrete Mathematics Given integers $n\ge s\ge 2$, let $e(n,s)$ stand for the maximum size of a family of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. The study of this quantity goes back to the 1960s, when Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all integer $m,s\ge 1$. The question of determining $e(n,s)$ is closely connected to its uniform counterpart, the subject of the famous Erdős Matching Conjecture. The problem of determining $e(n,s)$ has proven to be very hard and, in spite of some progress during these years, even a general conjecture concerning the value of $e(n,s)$ is missing. In this paper, we completely solve the problem for $n\le 3s$. In this regime, the average size of a set in an $s$-matching is at most $3$, and it is a delicate interplay between the `missing' $2$- and $3$-element sets that plays a key role here. Four types of extremal families appear in the characterization. Our result sheds light on how the extremal function $e(n,s)$ may behave in general. |
| title | A complete solution of the Erdős-Kleitman matching problem for $n\le 3s$ |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2511.21628 |