Fractional L-intersecting families

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{fractional $L$-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.