The Exact Erdős-Ko-Rado Theorem for 3-wise $t$-intersecting uniform families
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918379819892736 |
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| author | Frankl, Peter Wang, Jian |
| author_facet | Frankl, Peter Wang, Jian |
| contents | Let $\mathcal{F}$ be a family of $k$-element subsets of $\{1,2,\ldots,n\}$. For $t\geq 1$, we say that $\mathcal{F}$ is {\it 3-wise $t$-intersecting} if $|F_1\cap F_2\cap F_3|\geq t$ for all $F_1,F_2,F_3\in \mathcal{F}$. In the present paper, we prove that if $\mathcal{F}$ is 3-wise $t$-intersecting and $n\geq \frac{\sqrt{4t+9}-1}{2}k$, $k>t\geq 46$, then $|\mathcal{F}|\leq \binom{n-t}{k-t}$. The restriction on $n$ is asymptotically best possible. The corresponding result for non-trivial 3-wise $t$-intersecting families is obtained as well for $n\geq \frac{\sqrt{4t+9}-1}{2}k$ and $k>t\geq 55$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07991 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Exact Erdős-Ko-Rado Theorem for 3-wise $t$-intersecting uniform families Frankl, Peter Wang, Jian Combinatorics Let $\mathcal{F}$ be a family of $k$-element subsets of $\{1,2,\ldots,n\}$. For $t\geq 1$, we say that $\mathcal{F}$ is {\it 3-wise $t$-intersecting} if $|F_1\cap F_2\cap F_3|\geq t$ for all $F_1,F_2,F_3\in \mathcal{F}$. In the present paper, we prove that if $\mathcal{F}$ is 3-wise $t$-intersecting and $n\geq \frac{\sqrt{4t+9}-1}{2}k$, $k>t\geq 46$, then $|\mathcal{F}|\leq \binom{n-t}{k-t}$. The restriction on $n$ is asymptotically best possible. The corresponding result for non-trivial 3-wise $t$-intersecting families is obtained as well for $n\geq \frac{\sqrt{4t+9}-1}{2}k$ and $k>t\geq 55$. |
| title | The Exact Erdős-Ko-Rado Theorem for 3-wise $t$-intersecting uniform families |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.07991 |