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A Product Version of the Hilton-Milner-Frankl Theorem

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arxiv 2206.07217 v2 pith:S677YVW7 submitted 2022-06-15 math.CO

A Product Version of the Hilton-Milner-Frankl Theorem

classification math.CO
keywords mathcalproductcoloncrossfamilieshilton-milner-franklintersectingldots
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Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross $t$-intersecting if $|F\cap G|\geq t$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $|\cap \{F\colon F\in \mathcal{F}\}|<t$, $|\cap \{G\colon G\in\mathcal{G}\}|<t$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross $t$-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$ for $n\geq 4(t+2)^2k^2$, $k\geq 5$, which is a product version of the Hilton-Milner-Frankl Theorem.

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