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

Abstract

Let $L=\{\frac{a_1}{b_1},\dots ,\frac{a_s}{b_s}\}$, where for every $i\in [s]$, $\frac{a_i}{b_i}\in [0,1)$ is an irreducible fraction. Let $\mathcal{F}=\{A_1,\dots ,A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is a \emph{r-wise fractional $L$-intersecting family} if for every distinct $i_1,i_2,\dots ,i_r\in [m]$, there exists an $\frac{a}{b}\in L$ such that $|A_{i_1}\cap A_{i_2}\cap \dots \cap A_{i_r}|\in \{\frac{a}{b}|A_{i_1}|,\frac{a}{b}|A_{i_2}|,\dots ,\frac{a}{b}|A_{i_r}|\}$. 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].

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