Maximal intersecting families revisited

The well-known Erdős–Ko–Rado theorem states that for $n>2k$, every intersecting family of $k$-sets of $[n]:=\{1,\dots ,n\}$ has at most $\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)$ 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 $k\ge 1,t\ge 0$ and $n\ge 2k+t$ be integers. Frankl (2016) proved that if $\mathcal{F}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k+t}\right)$ and $\mathcal{G}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ are cross-intersecting families, and $\mathcal{F}$ is non-empty and $(t+1)$-intersecting, then $\mathrm{\mid }\mathcal{F}\mathrm{\mid }+\mathrm{\mid }\mathcal{G}\mathrm{\mid }\le \left(\genfrac{}{}{0px}{}{n}{k}\right)-\left(\genfrac{}{}{0px}{}{n-k-t}{k}\right)+1$. 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 Erdős–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.