$1$-cross intersecting set pair systems are small

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 $i\ne j$. 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 $|A_i|\le a$ and $|B_i|\le b$ for all $i$. We prove a conjecture of Füredi, Gyárfás, and Király [Combin. Probab. Comput. 32 (2023)] that asserts $m(n,n,1)/\left(\genfrac{}{}{0ex}{}{2n}{n}\right)\to 0$ as $n\to \infty$.

7 pages