Fractional cross intersecting families

Abstract

Let $\mathcal{A}=\{A_1,...,A_p\}$ and $\mathcal{B}=\{B_1,...,B_q\}$ be two families of subsets of $[n]$ such that for every $i\in [p]$ and $j\in [q]$, $|A_i\cap B_j|=\frac{c}{d}|B_j|$, where $\frac{c}{d}\in [0,1]$ is an irreducible fraction. We call such families "$\frac{c}{d}$-cross intersecting families". In this paper, we find a tight upper bound for the product $|\mathcal{A}||\mathcal{B}|$ and characterize the cases when this bound is achieved for $\frac{c}{d}=\frac{1}{2}$. Also, we find a tight upper bound on $|\mathcal{A}||\mathcal{B}|$ when $\mathcal{B}$ is $k$-uniform and characterize, for all $\frac{c}{d}$, the cases when this bound is achieved.

15 pages