Bollobás-type theorems for hemi-bundled two families. (February 2022)

• Record Type: Journal Article
• Authors: Yu, Wenjun; Kong, Xiangliang; Xi, Yuanxiao; Zhang, Xiande; Ge, Gennian
• Is Part Of: European journal of combinatorics. Volume 100(2022)
• Publication Date: 2022-02
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2021.103438
• Languages: English
• ISSNs: 0195-6698

Abstract

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.