Maximal Fractional Cross-Intersecting Families

Abstract

Given an irreducible fraction $\frac{c}{d} \in [0,1]$, a pair $(\mathcal {A},\mathcal {B})$ is called a $\frac{c}{d}$-cross-intersecting pair of $2^{[n]}$ if $\mathcal {A}, \mathcal {B}$ are two families of subsets of [n] such that for every pair $A \in \mathcal {A}$ and $B\in \mathcal {B}$, $|A \cap B|= \frac{c}{d}|B|$. Mathew et al. (Graphs Comb 37:471–484, 2019) proved that $|\mathcal {A}||\mathcal {B}|\le 2^n$ if $(\mathcal {A}, \mathcal {B})$ is a $\frac{c}{d}$-cross-intersecting pair of $2^{[n]}$ and characterized all the pairs $(\mathcal {A},\mathcal {B})$ with $|\mathcal {A}||\mathcal {B}|=2^n$, such a pair also is called a maximal $\frac{c}{d}$-cross-intersecting pair of $2^{[n]}$, when $\frac{c}{d}\in \{0,\frac{1}{2}, 1\}$. In this note, we characterize all the maximal $\frac{c}{d}$-cross-intersecting pairs $(\mathcal {A},\mathcal {B})$ when $0<\frac{c}{d}<1$ and $\frac{c}{d}\not =\frac{1}{2}$, this result answers a question proposed by Mathew et al. (2019).