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A semidefinite programming approach to cross 2-intersecting families
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A semidefinite programming approach to cross 2-intersecting families
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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.
Forward citations
Cited by 4 Pith papers
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Extremal cross $t$-intersecting families under $t$-covering number constraints for vector spaces
For cross t-intersecting families of k- and l-dimensional subspaces with no common t-subspace, the product-maximizing pairs are exactly three explicit families A, H, and C.
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On extremal cross $t$-intersecting families with $t$-covering number conditions
The paper characterizes extremal cross t-intersecting families maximizing |F1| |F2| under tau_t(F1), tau_t(F2) >= t+1 and describes maximal t-intersecting families with tau_t = t+1.
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On $\ell$-weakly cross $t$-intersecting families for sets and vector spaces
For n large enough, the product of the sizes of two ℓ-weakly cross t-intersecting families of k- and k'-dimensional subspaces is at most the product of two Gaussian binomial coefficients.
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On $\ell$-weakly cross $t$-intersecting families for sets and vector spaces
Alternative proof for the set case of the ℓ-weakly cross t-intersecting theorem plus an explicit product bound |F|·|G| ≤ Gaussian-binomial product for subspaces when n meets a stated lower bound.
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