On Sizes of 1-Cross Intersecting Set Pair Systems

Let $ \{(A_{i},B_{i})\}_{i=1}^{m} be a set pair system. Füredi, Gyárfás, and Király called it 1 -cross intersecting if |A_{i}\cap B_{j}| is when i\neq j 0 i=j . They studied the systems their generalizations and, in particular, considered m(a,b,1) , maximum size of system which |A_{i}|\leq |B_{i}|\leq b for all i proved that m(n,n,1)\geq 5^{(n-1)/2} asked whether there are upper bounds on m(n,n,1) significantly better than classical bound {2n\choose n} Bollobás cross systems. Answering one questions, Holzman recently a,b\geq 2 then m(a,b,1)\leq\frac{29}{30}\binom{a+b}{a} He also conjectured factor \frac{29}{30} his can replaced by \frac{5}{6} Our goal to prove this bound.