Bollobás-type theorems for hemi-bundled two families. (February 2022) - Yu, Wenjun; Kong, Xiangliang; Xi, Yuanxiao; Zhang, Xiande; Ge, Gennian
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 ≤ i ≤ m . Suppose that A i ∩ B j = 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 + b a . In this paper, we consider a variant of this problem by requiring 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 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 ( A i, B i ) i = 1 m for the primary case of the theorem for finite sets.