A well-known result of Bollobás 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 $m \le {a+b \choose a}$. Füredi, Gyárfás and Király 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.