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On the maximum number of distinct intersections in an intersecting family
arXiv - CS - Discrete Mathematics Pub Date : 2021-08-01 , DOI: arxiv-2108.00479
Peter Frankl, Sergei Kiselev, Andrey Kupavskii

For $n > 2k \geq 4$ we consider intersecting families $\mathcal F$ consisting of $k$-subsets of $\{1, 2, \ldots, n\}$. Let $\mathcal I(\mathcal F)$ denote the family of all distinct intersections $F \cap F'$, $F \neq F'$ and $F, F'\in \mathcal F$. Let $\mathcal A$ consist of the $k$-sets $A$ satisfying $|A \cap \{1, 2, 3\}| \geq 2$. We prove that for $n \geq 50 k^2$ $|\mathcal I(\mathcal F)|$ is maximized by $\mathcal A$.

中文翻译:

关于相交族中不同交点的最大数量

对于 $n > 2k \geq 4$,我们考虑由 $k$-$\{1, 2, \ldots, n\}$ 的子集组成的交叉族 $\mathcal F$。让 $\mathcal I(\mathcal F)$ 表示所有不同的交集 $F \cap F'$, $F \neq F'$ 和 $F, F'\in \mathcal F$ 的族。令 $\mathcal A$ 包含满足 $|A \cap \{1, 2, 3\}| 的 $k$-集合 $A$ \geq 2$。我们证明对于 $n \geq 50 k^2$ $|\mathcal I(\mathcal F)|$ 被 $\mathcal A$ 最大化。
更新日期:2021-08-03
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