Non-uniform pairwise cross t-intersecting families

2025/08/27 by Yongjiang Wu, Wu, Yongjiang, Yongtao Li +5
#math.CO

paper · pdf · doi:10.48550/arxiv.2508.19725

Abstract

Let n≥ t≥ 1 and A₁, A₂, …, A_m ⊆ 2^[n] be non-empty families. We say that they are pairwise cross t-intersecting if |A_i∩ A_j|≥ t holds for any A_i∈ A_i and A_j∈ A_j with i≠ j. In the case where m=2 and A₁=A₂, determining the maximum size M(n,t) of a non-uniform t-intersecting family of sets over [n] was solved by Katona (1964), and enhanced by Frankl (2017), and recently by Li and Wu (2024). In this paper, we establish the following upper bound: if A₁, A₂, …, A_m ⊆ 2^[n] are non-empty pairwise cross t-intersecting families, then ∑_i=1^m |A_i| ≤ max \ ∑_k=t ⁿ\binomnk + m - 1, m M(n, t) \. Furthermore, we provide a complete characterization of the extremal families that achieve the bound. Our result not only generalizes an old result of Katona (1964) for a single family, but also extends a theorem of Frankl and Wong (2021) for two families. Moreover, our result could be viewed as a non-uniform version of a recent theorem of Li and Zhang (2025). The key in our proof is to utilize the generating set method and the pushing-pulling method together.

Related

• The maximum sum of sizes of non-empty cross L-intersecting families
• A result for hemi-bundled cross-intersecting families
• The maximum sum of sizes of non-empty cross t-intersecting families
• Non-uniform Cross-intersecting Families
• Intersecting non-uniform families containing subfamilies