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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2605.08603 |
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| _version_ | 1866915996127723520 |
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| author | Frankl, Peter Wang, Jian |
| author_facet | Frankl, Peter Wang, Jian |
| contents | A family $\mathcal{F}\subset \binom{[n]}{k}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The covering number of a family $\mathcal{F}$ is defined as the minimum size of $T\subset [n]$ such that $T\cap F\neq \emptyset$ for all $F\in \mathcal{F}$. In 1980, the first author proved that for sufficiently large $n$, any intersecting $k$-graph $\mathcal{F}$ with covering number at least three, satisfies $|\mathcal{F}|\leq \binom{n-1}{k-1}-\binom{n-k}{k-1}-\binom{n-k-1}{k-1}+\binom{n-2k}{k-1}+\binom{n-k-2}{k-3}+3$. There was very little progress during more than forty years but recently (cf. \cite{FW25}) with a completely
different approach we proved the same result for the full range $n\geq 2k$ and $k\geq 7$. In this short paper we prove the same inequality for all the remaining cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intersecting families with covering number three II Frankl, Peter Wang, Jian Combinatorics A family $\mathcal{F}\subset \binom{[n]}{k}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The covering number of a family $\mathcal{F}$ is defined as the minimum size of $T\subset [n]$ such that $T\cap F\neq \emptyset$ for all $F\in \mathcal{F}$. In 1980, the first author proved that for sufficiently large $n$, any intersecting $k$-graph $\mathcal{F}$ with covering number at least three, satisfies $|\mathcal{F}|\leq \binom{n-1}{k-1}-\binom{n-k}{k-1}-\binom{n-k-1}{k-1}+\binom{n-2k}{k-1}+\binom{n-k-2}{k-3}+3$. There was very little progress during more than forty years but recently (cf. \cite{FW25}) with a completely different approach we proved the same result for the full range $n\geq 2k$ and $k\geq 7$. In this short paper we prove the same inequality for all the remaining cases. |
| title | Intersecting families with covering number three II |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.08603 |