$r$-wise fractional $L$-intersecting family

Abstract: 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 [Niranjan et.al, Fractional $L$-intersecting families, The Electronic Journal of Combinatorics, 2019].