Problems and results on 1-cross-intersecting set pair systems

Abstract The notion of cross-intersecting set pair system of size $m$ , $ (\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m )$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne \emptyset$ , was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $m\le\binom{a+b}{a}$ if $|A_i|\le a$ and $|B_i|\le b$ for each $i$ . Our central problem is to see how this bound changes with the additional condition $|A_i\cap B_j|=1$ for $i\ne j$ . Such a system is called $1$ -cross-intersecting. We show that these systems are related to perfect graphs, clique partitions of graphs, and finite geometries. We prove that their maximum size is at least $5^{n/2}$ for $n$ even, $a=b=n$ , equal to $\bigl (\lfloor \frac{n}{2}\rfloor +1\bigr )\bigl (\lceil \frac{n}{2}\rceil +1\bigr )$ if $a=2$ and $b=n\ge 4$ , at most $|\cup _{i=1}^m A_i|$ , asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ( $|A_i\cap A_j|\le 1$ for $i\ne j$ ), asymptotically ${1\over 2}n^2$ if $\{A_i\}$ and $\{B_i\}$ are both linear hypergraphs.