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

TL;DR

This paper introduces and studies r-wise fractional L-intersecting families, a generalization of fractional L-intersecting families, exploring their properties and extending previous combinatorial intersection concepts.

Contribution

It generalizes the concept of fractional L-intersecting families to r-wise intersections, providing new theoretical insights into their structure and properties.

Findings

Established bounds for the size of r-wise fractional L-intersecting families

Extended previous results on fractional L-intersecting families to r-wise cases

Provided new combinatorial characterizations and properties

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].