A unified approach to cross-intersection problems with applications to Hilton--Milner type theorems and stability
Abstract
We develop a new approach to cross-intersection problems in extremal set theory. The method builds on the iterative procedure introduced by Kupavskii and Zakharov (2024) and the t-cover method. It provides a flexible framework for deriving extremal and stability results for cross t-intersecting families. Our approach applies to a variety of combinatorial objects. As an application, we prove a product version of the seminal Erdős--Ko--Rado theorem for sufficiently spread set systems. Two families ℱ and 𝒢 of k-subsets of [n] are called cross t-intersecting if |F∩ G|≥ t for all F∈ℱ and G∈𝒢. We determine the families maximizing min{|ℱ|, |𝒢|} for large n and all t≥2, generalizing results of Mörs (1985) and Füredi (1995) for cross $1$-intersecting families. We then determine the families maximizing |ℱ||𝒢| under the condition max{|∩F∈ℱF|,|∩G∈𝒢G|}<t for large n. This improves the bound obtained by Frankl and Wang (2024), and provides a characterization of extremal configurations. For a family ℱ of subsets of [n], we introduce its t-diversity γt(ℱ), defined as the minimum number of sets from ℱ not containing a fixed t-subset. This serves as a natural generalization of the important notion of diversity for t=1. We obtain a stability result via γt, and determine the maximum of min{γt(ℱ),γt(𝒢)} for cross t-intersecting families ℱ and 𝒢. These yield new results for t-intersecting families, including a stability theorem towards a conjecture of Ellis, Keller and Lifshitz (2019), which may also be regarded as a t-intersection version, for large n, of an influential theorem of Frankl (1987).
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BibTeX
@misc{wen2026unified,
title = {{A unified approach to cross-intersection problems with applications to Hilton--Milner type theorems and stability}},
author = {Jie Wen and Benjian Lv},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.03315},
eprint = {2607.03315},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.03315},
}
RIS
TY - UNPB
TI - A unified approach to cross-intersection problems with applications to Hilton--Milner type theorems and stability
AU - Jie Wen
AU - Benjian Lv
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2607.03315
UR - https://arxiv.org/abs/2607.03315
AB - We develop a new approach to cross-intersection problems in extremal set theory. The method builds on the iterative procedure introduced by Kupavskii and Zakharov (2024) and the t-cover method. It provides a flexible framework for deriving extremal and stability results for cross t-intersecting families. Our approach applies to a variety of combinatorial objects. As an application, we prove a product version of the seminal Erdős--Ko--Rado theorem for sufficiently spread set systems. Two families ℱ and 𝒢 of k-subsets of [n] are called cross t-intersecting if |F∩ G|≥ t for all F∈ℱ and G∈𝒢. We determine the families maximizing min{|ℱ|, |𝒢|} for large n and all t≥2, generalizing results of Mörs (1985) and Füredi (1995) for cross $1$-intersecting families. We then determine the families maximizing |ℱ||𝒢| under the condition max{|∩_F∈ℱF|,|∩_G∈𝒢G|}<t for large n. This improves the bound obtained by Frankl and Wang (2024), and provides a characterization of extremal configurations. For a family ℱ of subsets of [n], we introduce its t-diversity γ_t(ℱ), defined as the minimum number of sets from ℱ not containing a fixed t-subset. This serves as a natural generalization of the important notion of diversity for t=1. We obtain a stability result via γ_t, and determine the maximum of min{γ_t(ℱ),γ_t(𝒢)} for cross t-intersecting families ℱ and 𝒢. These yield new results for t-intersecting families, including a stability theorem towards a conjecture of Ellis, Keller and Lifshitz (2019), which may also be regarded as a t-intersection version, for large n, of an influential theorem of Frankl (1987).
ER -