Abstract
Let {(Ai,Bi)}i=1m be a collection of pairs of sets with |Ai|=a and |Bi|=b for 1≤i≤m. Suppose that Ai∩Bj=0̸ if and only if i=j, then by the famous Two Families Theorem of Bollobás, we have the size of this collection m≤a+ba. In this paper, we consider a variant of this problem by requiring {Ai}i=1m to be intersecting additionally. Using exterior algebra method, we prove a weighted Bollobás-type theorem for finite dimensional real vector spaces under these constraints. As a consequence, we obtain a similar theorem for finite sets, which settles a recent conjecture of Gerbner et al. (2020). Moreover, we also determine the unique extremal structure of {(Ai,Bi)}i=1m for the primary case of the theorem for finite sets.
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