Fractional L-intersecting families

Let $L=\{\frac{a_1}{b_1},\dots ,\frac{a_s}{b_s}\}$, where for every $i \in [s]$,$\frac{a_i}{b_i} \in [0,1)$isanirreduciblefraction\mathrm{.}Let$$\mathcal{F}$ = {A_1, \ldots , A_m}$beafamilyofsubsetsof$[n]$\mathrm{.}Wesay$$\mathcal{F}$is a \emph{fractionalL$-intersecting family} if for every distinct $i,j\in [m]$, there exists an $\frac{a}{b}\in L$ such that $|A_i \cap A_j| \in { \frac{a}{b}|A_i|, \frac{a}{b} |A_j|}$. In this paper, we introduce and study the notion of fractional $L$-intersecting families.