A bound for 1-cross intersecting set pair systems

A well-known result of Bollobás says that if {(Ai,Bi)}i=1m is a set pair system such that |Ai|≤a and |Bi|≤b for 1≤i≤m, and Ai∩Bj≠0̸ if and only if i≠j, then m≤a+ba. Füredi, Gyárfás and Király recently initiated the study of such systems with the additional property that |Ai∩Bj|=1 for all i≠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.