Bollob\'as type theorems for hemi-bundled two families

Let $\{(A_i,B_i){\}}_{i=1}^m$ be a collection of pairs of sets with $\mathrm{\mid }{A}_i\mathrm{\mid }=a$ and $\mathrm{\mid }{B}_i\mathrm{\mid }=b$ for $1\le i\le m$. Suppose that $A_i\cap B_j=\mathrm{\varnothing }$ if and only if $i=j$, then by the famous Bollob'{a}s theorem, we have the size of this collection $m\le \left(\genfrac{}{}{0px}{}{a+b}{a}\right)$. In this paper, we consider a variant of this problem by setting $\{A_i{\}}_{i=1}^m$ to be intersecting additionally. Using exterior algebra method, we prove a weighted Bollob'{a}s type theorem for finite dimensional real vector spaces under these constraints. As a consequence, we have a similar theorem for finite sets, which settles a recent conjecture of Gerbner et. al \cite{GKMNPTX2019}. Moreover, we also determine the unique extremal structure of $\{(A_i,B_i){\}}_{i=1}^m$ for the primary case of the theorem for finite sets.