Király–Nagy–Pálvölgyi–Visontai conjecture on weakly cross intersecting set pair systems

Let $A_1,A_2,\dots,A_m$ and $B_1,B_2,\dots,B_m$ be sets such that $|A_i|=k$ and $|B_i|=l$ for all $1\le i\le m$. Suppose that $A_i\cap B_i=\emptyset$ for all $1\le i\le m$, and that $A_i\cap B_j\neq\emptyset$ or $A_j\cap B_i\neq\emptyset$ for all distinct $i,j$. Such a system is called a $(k,l)$-weakly cross intersecting set pair system, and let $m_{\max}(k,l)$ denote the largest $m\in\mathbb{Z}$ for which one exists.

Király–Nagy–Pálvölgyi–Visontai conjecture.

$$m_{\max}(k,l)\leq 2\binom{k+l}{k}.$$

Tuza's upper bound and the cited construction show that the conjectured bound is asymptotically sharp up to a factor approaching $1$. The conjecture concerns the maximum size of weakly cross intersecting set pair systems.