A semidefinite programming approach to cross $2$-intersecting families
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910883331964928 |
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| author | Tanaka, Hajime Tokushige, Norihide |
| author_facet | Tanaka, Hajime Tokushige, Norihide |
| contents | Let $k\geq 2$ and $n\geq 3(k-1)$. Let $\mathcal{F}$ and $\mathcal{G}$ be families of $k$-element subsets of an $n$-element set. Suppose that $|F\cap G|\geq 2$ for all $F\in\mathcal{F}$ and $G\in\mathcal{G}$. We show that $|\mathcal{F}||\mathcal{G}|\leq\binom{n-2}{k-2}^2$, and determine the extremal configurations. This settles the last unsolved case of a recent result by Zhang and Wu (J. Combin. Theory Ser. B, 2025). We also obtain the corresponding result in the product measure setting. Our proof is done by solving semidefinite programming problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14844 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A semidefinite programming approach to cross $2$-intersecting families Tanaka, Hajime Tokushige, Norihide Combinatorics Let $k\geq 2$ and $n\geq 3(k-1)$. Let $\mathcal{F}$ and $\mathcal{G}$ be families of $k$-element subsets of an $n$-element set. Suppose that $|F\cap G|\geq 2$ for all $F\in\mathcal{F}$ and $G\in\mathcal{G}$. We show that $|\mathcal{F}||\mathcal{G}|\leq\binom{n-2}{k-2}^2$, and determine the extremal configurations. This settles the last unsolved case of a recent result by Zhang and Wu (J. Combin. Theory Ser. B, 2025). We also obtain the corresponding result in the product measure setting. Our proof is done by solving semidefinite programming problems. |
| title | A semidefinite programming approach to cross $2$-intersecting families |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.14844 |