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 $1\le i\le m$. Suppose that $A_i\cap B_j=\varnothing$ if and only if $i=j$, then by the famous Bollobás theorem, we have the size of this collection $m\le \left(\genfrac{}{}{0ex}{}{a+b}{a}\right)$. 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 Bollobás 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.

References in corpus (2)

• Set systems related to a house allocation problem
• Combinatorics in the exterior algebra and the Bollobás Two Families Theorem