Maximal non-empty cross s -union families
Year: 2025
Type: article
Abstract: Two families of sets F and G are said to be cross s -union if for any F ∈ F and G ∈ G , | F ∪ G | ≤ s . In 2021, Frankl and Wong proved that if F , G ⊆ 2 [ n ] are non-empty cross s -union, then | F | + | G | ≤ ∑ i = 0 s n i + 1 . Moreover, for s < n − 1 , equality holds if and only if F , G = { 0̸ } , { G ⊆ [ n ] : | G | ≤ s } . In this paper, we give a new method to prove this result. Our method also allows us to establish a vector space version and a hereditary family extension. As a byproduct, we revisit the vector space version of the Katona s -union theorem due to Frankl and Tokushige, and characterize the extremal families for the case s = n − 1 .
Source: Discrete Applied Mathematics
Authors: Yongjiang Wu, Yongtao Li, Zhiyi Liu, Lihua Feng, Tingzeng Wu
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W7116068969