论文标题
在层次上封闭的分数相交家庭
On hierarchically closed fractional intersecting families
论文作者
论文摘要
对于一组正正确分数和正整数$ r \ geq 2 $，部分$ r $ r $ claped $ l $ -l $ - 划分家庭是一个集合$ \ nathcal {f} \ subset \ subset \ subset \ subset \ mathcal {p}（p}（[n]）$，与任何$ 2 \ leq leq t \ leq l leq l leq l leq r $ a_1 c。 \ Mathcal {f} $中存在L $中的$θ\，以便$ \ lvert a_1 \ cap \ dotsb \ cap a_t \ rvert \ in \ in \ {θ\ lvert a_1 \ rvert，\ rvert，\ dotsc，\ dotsc，fertc，θ\ lvert a_t a_t a _t \ rvert \ rvert \ rvert \ rvert \ \ \ \} $。在本文中，我们表明，对于$ r \ geq 3 $和$ l = \ {θ\} $任何分数$ r $ claped的$θ$ - 启用家庭最多都具有$ n $的尺寸，这是最不变的因素。我们还表明，在这种情况下，$θ= 1/2 $，我们有一个紧密的上限，$ \ lfloor \ frac {3n} {2} {2} \ rfloor-2 $，并且最大$ r $ r $ claped $（1/2）$ - 确定互联家庭的互联家庭被确定为同性恋。
For a set $L$ of positive proper fractions and a positive integer $r \geq 2$, a fractional $r$-closed $L$-intersecting family is a collection $\mathcal{F} \subset \mathcal{P}([n])$ with the property that for any $2 \leq t \leq r$ and $A_1, \dotsc, A_t \in \mathcal{F}$ there exists $θ\in L$ such that $\lvert A_1 \cap \dotsb \cap A_t \rvert \in \{ θ\lvert A_1 \rvert, \dotsc, θ\lvert A_t \rvert\}$. In this paper we show that for $r \geq 3$ and $L = \{θ\}$ any fractional $r$-closed $θ$-intersecting family has size at most linear in $n$, and this is best possible up to a constant factor. We also show that in the case $θ= 1/2$ we have a tight upper bound of $\lfloor \frac{3n}{2} \rfloor - 2$ and that a maximal $r$-closed $(1/2)$-intersecting family is determined uniquely up to isomorphism.
