Structure of large t-intersecting families I: Stability for the Hilton--Milner--Frankl theorem
Abstract
We study the structure of large t-intersecting families. A family of k-subsets of an n-set is t-intersecting if every two of its members intersect in at least t elements. A t-intersecting family is non-trivial if no t-subset is contained in all its members. We prove several stability results for the seminal Hilton--Milner--Frankl theorem. First, for any fixed η,ε,θ∈(0,1), we prove that if k/t≥1+η and n=Ω(tk1+ε), then every non-trivial t-intersecting family of size greater than (1+θ)|𝒦| is a subfamily of one of the two extremal families in the theorem, where 𝒦 is an explicit large non-trivial t-intersecting family. The key ingredient in the proof is a removal lemma. We also obtain a classification of all t-intersecting families with size bounded below by |𝒦| minus an explicit lower-order term, provided that k≥ t+4≥6 and n≥ t+6·max{(t+2)2, k(k−t)}. This strengthens results of Cao--Lv--Wang (2021) and Frankl (2025) for a broad range of k and t (for example, when k−t≥2√t). As an application of this classification, we determine the largest t-intersecting families for each prescribed lower bound on t-diversity not exceeding t(n−k), thereby obtaining t-intersection versions of results of Han and Kohayakawa (2017) and Kupavskii (2025). To establish these results, we develop techniques based on the spread approximation method and the t-cover method, which may be useful for other intersection problems.
Related
- A product theorem for r-cross intersecting families of subspaces
- Sharp stability for cross t-intersecting families of permutations in the linear range
- The Hilton-Milner theorem in the ℓ2-norm
- Three-Layer Intersecting Temperate Families
- A product inequality and its application to cross-intersecting families
BibTeX
@misc{wen2026structure,
title = {{Structure of large $t$-intersecting families I: Stability for the Hilton--Milner--Frankl theorem}},
author = {Jie Wen and Benjian Lv},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.14197},
eprint = {2608.14197},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14197},
}
RIS
TY - UNPB
TI - Structure of large t-intersecting families I: Stability for the Hilton--Milner--Frankl theorem
AU - Jie Wen
AU - Benjian Lv
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.14197
UR - https://arxiv.org/abs/2608.14197
AB - We study the structure of large t-intersecting families. A family of k-subsets of an n-set is t-intersecting if every two of its members intersect in at least t elements. A t-intersecting family is non-trivial if no t-subset is contained in all its members. We prove several stability results for the seminal Hilton--Milner--Frankl theorem. First, for any fixed η,ε,θ∈(0,1), we prove that if k/t≥1+η and n=Ω(tk^1+ε), then every non-trivial t-intersecting family of size greater than (1+θ)|𝒦| is a subfamily of one of the two extremal families in the theorem, where 𝒦 is an explicit large non-trivial t-intersecting family. The key ingredient in the proof is a removal lemma. We also obtain a classification of all t-intersecting families with size bounded below by |𝒦| minus an explicit lower-order term, provided that k≥ t+4≥6 and n≥ t+6·max{(t+2)^2, k(k−t)}. This strengthens results of Cao--Lv--Wang (2021) and Frankl (2025) for a broad range of k and t (for example, when k−t≥2√t). As an application of this classification, we determine the largest t-intersecting families for each prescribed lower bound on t-diversity not exceeding t(n−k), thereby obtaining t-intersection versions of results of Han and Kohayakawa (2017) and Kupavskii (2025). To establish these results, we develop techniques based on the spread approximation method and the t-cover method, which may be useful for other intersection problems.
ER -