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 $\frac cd$-cross-intersecting pair of $2^{ [n]}$, when $\frac cd\in\{ 0,\frac12, 1\}$. In this note, we characterize all the maximal $\frac cd$-cross-intersecting pairs $(\mathcal{ A},\mathcal{ B})$ when $0<\frac{ c}{ d}<1$ and $\frac cd\not=\frac 12$, this result answers a question proposed by Mathew, Ray, and Srivastava (2019).
Categories: math.CO