论文标题
$ r $ - 十字$ t $ - 通过必要的交叉点来建立家庭
$r$-cross $t$-intersecting families via necessary intersection points
论文作者
论文摘要
给定整数$ r \ geq 2 $和$ n，t \ geq 1 $我们称为tame $ \ mathcal {f} _1，\ dots，\ Mathcal {f} $ f_i \ in \ Mathcal {f} _i $，$ i \ in [r] $，我们有$ \ vert \ vert \ bigcap_ {i \ in [r]} f_i \ vert \ vert \ geq t $。我们对经典的希尔顿 - 米勒纳定理进行了强烈的概括。特别是，我们确定$ \ sum_ {j \ in [r]} \ vert \ mathcal {f} _j \ vert $ for $ r $ -cross $ -cross $ t $ -t $ - 在这些情况下为$ k $ -k $ -riniform-rioncrip the the the ty $ k $ - 统一的家庭或$ \ mathscr的任意亚formires of $ k $ rionight of。以前仅证明了这些结果的一些特殊情况。作为更普遍的结果，我们将获得上述定理，该实例考虑了$ r $ r $ t $ t $ ipterting家庭的措施。这还提供了$ \ sum_ {j \ in [r]} \ vert \ mathcal {f} _j \ vert $的最大值，可能是混合均匀性的家庭$ k_1，\ ldots，k_r $。
Given integers $r\geq 2$ and $n,t\geq 1$ we call families $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq\mathscr{P}([n])$ $r$-cross $t$-intersecting if for all $F_i\in\mathcal{F}_i$, $i\in[r]$, we have $\vert\bigcap_{i\in[r]}F_i\vert\geq t$. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross intersecting families. In particular, we determine the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for $r$-cross $t$-intersecting families in the cases when these are $k$-uniform families or arbitrary subfamilies of $\mathscr{P}([n])$. Only some special cases of these results had been proved before. We obtain the aforementioned theorems as instances of a more general result that considers measures of $r$-cross $t$-intersecting families. This also provides the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for families of possibly mixed uniformities $k_1,\ldots,k_r$.
