On sizes of 1-cross intersecting set pair systems

Let $\{(A_i,B_i){\}}_{i=1}^m$ be a set pair system. F"{u}redi, Gy'{a}rf'{a}s and Kir'{a}ly called it {\em $1$-cross intersecting} if $\mathrm{\mid }{A}_i\cap B_j\mathrm{\mid }$ is $1$ when $i\mathrm{\ne }j$ and $0$ if $i=j$. They studied such systems and their generalizations, and in particular considered $m(a,b,1)$ – the maximum size of a $1$-cross intersecting set pair system in which $\mathrm{\mid }{A}_i\mathrm{\mid }\le a$ and $|B_i|\leq b$forall$i$\mathrm{.}F\ddot{u}redi,Gy\acute{a}rf\acute{a}sandKir\acute{a}lyprovedthat$m(n,n,1)\geq 5^{(n-1)/2}$andaskedwhetherthereareupperboundson$m(n,n,1)$significantlybetterthantheclassicalbound${2n\choose n}$ of Bollob' as for cross intersecting set pair systems. Answering one of their questions, Holzman recently proved that if $a,b\geq 2$,then$m(a,b,1)\leq \frac{29}{30}\binom{a+b}{a}$. He also conjectured that the factor $\frac{29}{30}$ in his bound can be replaced by $\frac{5}{6}$. The goal of this paper is to prove this bound.