  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 $|A_i\cap B_j|$ is $1$
when $i\neq j$ 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 $|A_i|\leq a$ and $|B_i|\leq
b$ for all $i$. F\"{u}redi, Gy\'{a}rf\'{a}s and Kir\'{a}ly proved that
$m(n,n,1)\geq 5^{(n-1)/2}$ and asked whether there are upper bounds on
$m(n,n,1)$ significantly better than the classical bound ${2n\choose n}$ 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.
