  A well-known result of Bollob\'as says that if $\{(A_i, B_i)\}_{i=1}^m$ is a
set pair system such that $|A_i| \le a$ and $|B_i| \le b$ for $1 \le i \le m$,
and $A_i \cap B_j \ne \emptyset$ if and only if $i \ne j$, then $m \le {a+b
\choose a}$. F\"uredi, Gy\'arf\'as and Kir\'aly recently initiated the study of
such systems with the additional property that $|A_i \cap B_j| = 1$ for all $i
\ne j$. Confirming a conjecture of theirs, we show that this extra condition
allows an improvement of the upper bound (at least) by a constant factor.
