Bollobás type theorems for hemi-bundled two families

Abstract: Let ${(A_i,B_i)}{i=1}^{m}$ be a collection of pairs of sets with $|A_i|=a$ and $|B_i|=b$ for $1\leq i\leq m$. Suppose that $A_i\cap B_j=\emptyset$ if and only if $i=j$, then by the famous Bollob\'{a}s theorem, we have the size of this collection $m\leq {a+b\choose a}$. 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.