Structure of non-trivial intersecting families
Andrey Kupavskii
TL;DR
The paper tackles the problem of classifying large intersecting families of $k$-subsets of an $n$-element set, extending classic results of Erdős–Ko–Rado, Hilton–Milner, and subsequent generalizations by Han–Kohayakawa and Kostochka–Mubayi. It develops a broad, sharp framework that describes the structure of all large intersecting $k$-uniform families without imposing extremely large $n$, using a combination of shifting and a bipartite switching technique to derive essentially optimality and uniqueness conclusions. The main result characterizes such families in terms of a maximal, inclusion-based construction $\mathcal{F}'$ built from a minimal subfamily $\mathcal{M}\subset \mathcal{F}(\bar{1})$, proving that $|\mathcal{F}|\le |\mathcal{F}'|$ with equality typically forcing isomorphism to $\mathcal{F}'$ (for $k\ge 5$), and treating $k=4$ with explicit exceptional cases; it also yields bounds like $|\mathcal{F}|\le |\mathcal{J}_{k-t+1}|$ under stronger intersection constraints. The approach harmonizes shifting, Kruskal–Katona shadow analysis, and cross-intersection bounds (KZ) to deliver a comprehensive extremal classification applicable to a wide range of parameters, thereby unifying and sharpening prior results.
Abstract
We say that a family of $k$-subsets of an $n$-element set is {\it intersecting}, if any two of its sets intersect. In this paper, we study the structure of large intersecting families. Several years ago, Han and Kohayakawa (Proc. AMS, 2017), and then Kostochka and Mubayi (Proc. AMS, 2017) obtained certain structural results concerning large intersecting families. In this paper, we extend and generalize their results, giving them a conclusive form.