On the size of an $r$-wise fractional $L$-intersecting family

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 an $r$-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 [$\href{https://mathscinet.ams.org/mathscinet-getitem?mr=3982269}{17}$].

Keywords

L-intersecting families, Fractional L-intersecting families

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Received 1 December 2021

Accepted 22 September 2022

Published 7 November 2023