  The notion of cross intersecting set pair system of size $m$,
$\Big(\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m\Big)$ with $A_i\cap B_i=\emptyset$ and
$A_i\cap B_j\ne\emptyset$, was introduced by Bollob\'as and it became an
important tool of extremal combinatorics. His classical result states that
$m\le {a+b\choose a}$ 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 the maximum size of a $1$-cross intersecting set
pair system 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.
