$1$-cross intersecting set pair systems are small

Source abstract

A set pair system $\{(A_i,B_i)\}_{i=1}^m$ is $1$-cross intersecting if $|A_i \cap B_i|=0$ for all $i$ and $|A_i \cap B_j| = 1$ whenever $i \neq j$. Let $m(a,b,1)$ denote the maximum size $m$ of a $1$-cross intersecting set pair system $\{(A_i,B_i)\}_{i=1}^m$ where $|A_i| \leq a$ and $|B_i| \leq b$ for all $i$. We prove a conjecture of Füredi, Gyárfás, and Király [Combin. Probab. Comput. 32 (2023)] that asserts $m(n,n,1)/\binom{2n}{n} \to 0$ as $n \to \infty$.