Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is a \emph{r-wise fractional $L$-intersecting family} if for every distinct $i_1,i_2, \ldots,i_r \in [m]$, there exists an $\frac{a}{b} \in L$ such that $|A_{i_1} \cap A_{i_2} \cap \ldots \cap A_{i_r}| \in \{ \frac{a}{b}|A_{i_1}|, \frac{a}{b} |A_{i_2}|,\ldots, \frac{a}{b} |A_{i_r}| \}$. In this paper, we introduce and study the notion of r-wise fractional $L$-intersecting families. This is a generalization of notion of fractional $L$-intersecting families studied in \cite{niranj2019}.

中文翻译：

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$r$-wise 分数 $L$-相交族

令 $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$，其中对于每个 $i \in [s]$，$\frac{a_i}{b_i } \in [0,1)$ 是一个不可约的分数。令 $\mathcal{F} = \{A_1, \ldots , A_m\}$ 是 $[n]$ 的子集族。如果对于每个不同的 $i_1,i_2, \ldots,i_r \in [m]$，都存在 $\ frac{a}{b} \in L$ 使得 $|A_{i_1} \cap A_{i_2} \cap \ldots \cap A_{i_r}| \in \{ \frac{a}{b}|A_{i_1}|, \frac{a}{b} |A_{i_2}|,\ldots, \frac{a}{b} |A_{i_r} | \}$。在本文中，我们介绍并研究了 r-wise 分数 $L$-相交族的概念。这是 \cite{niranj2019} 中研究的分数 $L$ 相交家庭概念的概括。