论文标题
相交家族的不同相交数量的阈值提高了阈值
An improved threshold for the number of distinct intersections of intersecting families
论文作者
论文摘要
$ \ {1,2，\ ldots，n \} $的子集的家庭$ \ Mathcal {f} $，如果$ | f \ cap g | \ geq t $对于任何两个成员$ f，g \ in \ mathcal {f} $以及一些正整数$ t $。如果$ t = 1 $，那么我们将家庭$ \ Mathcal {f} $称为相交。定义集合$ \ MATHCAL {i}（\ MATHCAL {f}）= \ {f \ cap g：f，g \ in \ Mathcal {f} \ text {and} f \ neq g \ \} $，以作为$ \ mathcal {f} $的所有独特的交集。弗兰克等人。事实证明，与$ \ Mathcal {i}（\ Mathcal {f}）$相交家庭的$ \ Mathcal {i} $ $ \ MATHCAL {f} $ $ k $ -subsets $ \ \ {1,2，\ ldots，n \} $的上限。他们的定理适用于整数$ n \ geq 50 k^2 $。在本文中，我们证明了$ \ mathcal {i}（\ Mathcal {f}）$ $ t $ tub-intersecting family $ \ mathcal {f} $的上限，前提是$ n $超过一定数量$ f（k，t）$。在此过程中，我们还改善了相交家庭的阈值$ k^2 $至$ k^{3/2+o（1）} $。
A family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ is called a $t$-intersecting family if $|F\cap G| \geq t$ for any two members $F, G \in \mathcal{F}$ and for some positive integer $t$. If $t=1$, then we call the family $\mathcal{F}$ to be intersecting. Define the set $\mathcal{I}(\mathcal{F}) = \{F\cap G: F, G \in \mathcal{F} \text{ and } F \neq G\}$ to be the collection of all distinct intersections of $\mathcal{F}$. Frankl et al. proved an upper bound for the size of $\mathcal{I}(\mathcal{F})$ of intersecting families $\mathcal{F}$ of $k$-subsets of $\{1,2,\ldots,n\}$. Their theorem holds for integers $n \geq 50 k^2$. In this article, we prove an upper bound for the size of $\mathcal{I}(\mathcal{F})$ of $t$-intersecting families $\mathcal{F}$, provided that $n$ exceeds a certain number $f(k,t)$. Along the way we also improve the threshold $k^2$ to $k^{3/2+o(1)}$ for the intersecting families.
