Maximal intersecting families revisited

TL;DR

This paper extends stability results for intersecting families, generalizes previous theorems, and provides new, simpler proofs for classical results like the Hilton--Milner theorem, enhancing understanding of combinatorial set systems.

Contribution

It introduces a new stability variant that unifies and extends prior results, and offers more straightforward proofs of key theorems in intersecting family theory.

Findings

Established a generalized stability result for cross-intersecting families.

Provided simplified proofs for Hilton--Milner, Han--Kohayakawa, and Huang--Peng theorems.

Demonstrated applications of cross-intersecting family results to classical stability theorems.

Abstract

The well-known Erd\H{o}s--Ko--Rado theorem states that for $n> 2k$, every intersecting family of $k$-sets of $[n]:=\{1,\ldots ,n\}$ has at most $ {n-1 \choose k-1}$ sets, and the extremal family consists of all $k$-sets containing a fixed element (called a full star). The Hilton--Milner theorem provides a stability result by determining the maximum size of a uniform intersecting family that is not a subfamily of a full star. The further stabilities were studied by Han and Kohayakawa (2017) and Huang and Peng (2024). Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. Let $k \geq 1, t\ge 0$ and $n \geq 2 k+t$ be integers. Frankl (2016) proved that if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, and $\mathcal{F}$ is non-empty and $(t+1)$-intersecting, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. Recently, Wu (2023) sharpened Frankl's result by establishing a stability variant. The aim of this paper is two-fold. Inspired by the above results, we first prove a further stability variant that generalizes both Frankl's result and Wu's result. Secondly, as an interesting application, we illustrate that the aforementioned results on cross-intersecting families could be used to establish the stability results of the Erd\H{o}s--Ko--Rado theorem. More precisely, we present new short proofs of the Hilton--Milner theorem, the Han--Kohayakawa theorem and the Huang--Peng theorem. Our arguments are more straightforward, and it may be of independent interest.