Abstract: Let $\{(A_i,B_i)\}_{i=1}^m$ be a set pair system. Furedi, Gyarfas and Kiraly called it $1$-cross intersecting if the size of intersection of $A_i$ and $B_j$ is $1$ when $i$ and $j$ are distinct, 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 $i$.
Answering one of their questions, Holzman recently proved that if $a,b\geq 2$, then $m(a,b,1)\leq (29/30)((a+b) \text{ choose } a)$. He also conjectured that the factor $29/30$ in his bound can be replaced by $5/6$. The goal of this talk is to sketch the proof of this improved bound.
This is joint work with Grace Mc.Court and Alexandr V Kostochka.
For Zoom information, please contact Sean at SEnglish (at) illinois (dot) edu.