On a conjecture of Tokushige for cross-$t$-intersecting families
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916461029621760 |
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| author | Zhang, Huajun Wu, Biao |
| author_facet | Zhang, Huajun Wu, Biao |
| contents | Two families of sets $\mathcal{A}$ and $\mathcal{B}$ are called cross-$t$-intersecting if $|A\cap B|\ge t$ for all $A\in \mathcal{A}$, $B\in \mathcal{B}$. An active problem in extremal set theory is to determine the maximum product of sizes of cross-$t$-intersecting families. This incorporates the classical Erdős--Ko--Rado (EKR) problem. In the present paper, we prove that if $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting families of $\binom {[n]}k$ with $k\ge t\ge 3$ and $n\ge (t+1)(k-t+1)$, then $|\mathcal{A}||\mathcal{B}|\le {\binom{n-t}{k-t}}^2$; moreover, if $n>(t+1)(k-t+1)$, then equality holds if and only if $\mathcal{A}=\mathcal{B}$ is a maximum $t$-intersecting subfamily of $\binom{[n]}{k}$. This confirms a conjecture of Tokushige for $t\ge 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22792 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a conjecture of Tokushige for cross-$t$-intersecting families Zhang, Huajun Wu, Biao Combinatorics Two families of sets $\mathcal{A}$ and $\mathcal{B}$ are called cross-$t$-intersecting if $|A\cap B|\ge t$ for all $A\in \mathcal{A}$, $B\in \mathcal{B}$. An active problem in extremal set theory is to determine the maximum product of sizes of cross-$t$-intersecting families. This incorporates the classical Erdős--Ko--Rado (EKR) problem. In the present paper, we prove that if $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting families of $\binom {[n]}k$ with $k\ge t\ge 3$ and $n\ge (t+1)(k-t+1)$, then $|\mathcal{A}||\mathcal{B}|\le {\binom{n-t}{k-t}}^2$; moreover, if $n>(t+1)(k-t+1)$, then equality holds if and only if $\mathcal{A}=\mathcal{B}$ is a maximum $t$-intersecting subfamily of $\binom{[n]}{k}$. This confirms a conjecture of Tokushige for $t\ge 3$. |
| title | On a conjecture of Tokushige for cross-$t$-intersecting families |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.22792 |