Problems and results on 1-cross intersecting set pair systems

The notion of cross intersecting set pair system of size $m$, $(\{A_i{\}}_{i=1}^m,\{B_i{\}}_{i=1}^m)$ with $A_i\cap B_i=\mathrm{\varnothing }$ and $A_i\cap B_j\mathrm{\ne }\mathrm{\varnothing }$, was introduced by Bollob'as and it became an important tool of extremal combinatorics. His classical result states that $m\le \left(\genfrac{}{}{0px}{}{a+b}{a}\right)$ if $\mathrm{\mid }{A}_i\mathrm{\mid }\le a$ and $\mathrm{\mid }{B}_i\mathrm{\mid }\le b$ for each $i$. Our central problem is to see how this bound changes with the additional condition $\mathrm{\mid }{A}_i\cap B_j\mathrm{\mid }=1$ for $i\mathrm{\ne }j$. Such a system is called $1$-cross intersecting. We show that the maximum size of a $1$-cross intersecting set pair system is – at least $5^{n\mathrm{/}2}$ for $n$ even, $a=b=n$, – equal to $(\lfloor \frac{n}{2}\rfloor +1)(\lceil \frac{n}{2}\rceil +1)$ if $a=2$ and $b=n\ge 4$, – at most $\mathrm{\mid }{\cup }_{i=1}^m{A}_i\mathrm{\mid }$, – asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ($|A_i\cap A_j|\le 1$for$i\ne j$), – asymptotically $\frac{1}{2}{n}^2$ if $\{A_i\}$ and $\{B_i\}$ are both linear hypergraphs.