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, Ray, and Srivastava [{\it\small Fractional cross intersecting families, Graphs and Comb., 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 $\frac{c}{d}\ne \frac{1}{2}$, this result answers a question proposed by Mathew, Ray, and Srivastava (2019).

7 pages