Fractional L-intersecting families

TL;DR

This paper introduces and investigates fractional L-intersecting families, a new combinatorial concept where intersections of set pairs relate proportionally to their sizes based on a specified set of fractions.

Contribution

The paper defines fractional L-intersecting families and explores their properties, establishing foundational results for this new class of combinatorial structures.

Findings

Characterization of fractional L-intersecting families

Bounds on the size of such families

Connections to existing intersection theorems

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.